Math 018 Review Sheet v.3


 Leo Kennedy
 1 years ago
 Views:
Transcription
1 Math 018 Review Sheet v.3 Tyrone Crisp Spring Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right. Negative slope the line slopes down to the right. Slope equals zero horizontal line. Slope undefined vertical line. Large slope (in absolute value) steep line. Eg, a line with slope 5 is steeper than a line with slope 1. A line with slope 6 is steeper than a line with slope. Equations: Slopeintercept form: The equation of the line with slope m and yintercept b is y mx+b. PointSlope form: The equation of the line through the point (x 1, y 1 ) and with slope m is y y 1 m(x x 1 ). Line through two points: to find the equation of the line through points (x 1, y 1 ) and (x, y ), first find the slope using the slope formula, then apply the pointslope formula to find the equation. Horizontal lines: The equation of a horizontal line has the form y b, where b is the yintercept. Vertical lines: The equation of a vertical line has the form x a, where a is the xintercept. Parallel and Perpendicular lines: If two lines are parallel, they have the same slope (or they are both vertical). If two lines are perpendicular, the product of their slopes is equal to 1 (or one is horizontal and the other vertical).
2 The condition for perpendicular lines can be rephrased as If two lines are perpendicular, then their slopes are negative reciprocals of one another. (To find the negative reciprocal of a number, write it as a fraction, turn the fraction upsidedown and multiply by 1.) Graphs: To find the yintercept of a line, put x 0 in its equation and solve for y. To find the xintercept, put y 0 in its equation and solve for x. To draw the graph of a line, find its x and yintercepts, plot these two points on the plane, and draw a straight line through them. If the x and yintercepts are both zero, you ll need to find another point before you can draw the line. Choose any value for x (1 is usually a good choice) and plug that value into the equation. Solve for y. This tells you that the point (x 1, y 1 ) lies on the line, where x 1 is the x value you chose, and y 1 is the resulting y value. Plot this point, and join the dots. An equation like y b gives a horizontal line; to graph it, just draw a horizontal line with yintercept b. An equation like x a gives a vertical line; to graph it, just draw a vertical line with xintercept a Linear Functions and Applications Linear Functions: If y and x are two quantities, then saying that y is a linear function of x means that the value of y depends entirely on the value of x, and that when you draw a graph of y against x you get a straight line. In this situation, we call y the dependent variable (because it depends on x). We call x the independent variable. To emphasise that a certain quantity is a function of x, we often write it as f(x) rather than y. Given a formula for a function f(x), you can find (for example) f(4) by substituting 4 for x in the formula. To find x such that f(x) 7 (for example), replace f(x) by 7 in the given formula and solve for x. Cost and BreakEven Analysis:
3 The cost of producing any product can be broken down into the fixed cost (eg. for equipping a factory, training staff, etc.) and a marginal cost (the cost in materials and labour required to produce one unit of the product). The cost function for producing x units of a certain product is given by the formula C(x) mx + b, where m is the marginal cost and b is the fixed cost. If the product is sold at a price of p dollars per unit, then the revenue function for selling x units is R(x) px. The profit function for producing and selling x units of a product is given by the formula P(x) R(x) C(x): Profit Revenue Cost. Negative profit means loss. The breakeven point is the number of units one would need to produce and sell in order to make zero profit. To find this point, find the formula for the profit function P(x) and then solve P(x) 0. Temperature: We knew that 0 C 3 F and 100 C 1 F. We then found a formula for converting between Celsius and Fahrenheit by finding the equation of the straight line through (0, 3) and (100, 1). The actual formula was F 9 C+3 (but you won t need to commit this equation to memory 5 for the exam). To convert from one temperature scale to the other, plug in the temperature you have for the appropriate variable, and solve for the other (eg. to convert 40 C into Fahrenheit, plug in C 40 and solve for F)..1  Systems of Linear Equations  The Echelon Method Linear Equations: A linear equation in the variables x 1, x,..., x n is an equation which can be written in the form a 1 x 1 + a x a n x n k, where a 1, a,..., a n, k are numbers. A solution to such an equation is a list (s 1, s,..., s n ) of numbers which make the equation correct when you substitute x 1 s, x s,..., x n s n into the equation. A system of linear equations is a collection of linear equations in the same variables. A solution to a system of linear equations is a solution which works for every equation in the system.
4 The above four points are just terminology. You don t need to memorise it wordforword, just understand what it means. There are three possibilities for the number of solutions to a given system: 1. Exactly one solution ( Unique Solution );. No solutions ( Inconsistent ); 3. Infinitely many solutions ( Dependent ). In the case of systems of two equations in two variables, you can determine which of the above possibilities applies by graphing the two lines and seeing whether they intersect at a single point (unique solution), not at all (inconsistent), or at infinitely many points, i.e. they are the same line (dependent). Solving Linear Systems  Echelon Method: The following transformations can be applied to a system without changing its solutions: 1. Interchange two equations.. Multiply one equation by a nonzero constant. 3. Replace one equation by a nonzero multiple of itself plus a multiple of another equation. The strategy for solving a system of two equations in two variables is as follows: 1. Eliminate the first variable (usually called x) from the second equation.. Solve the second equation for the second variable (usually called y). 3. Substitute this yvalue into the first equation and solve for x. If you get an equation like 0 b, where b is some nonzero number, then the system is inconsistent, and there is nothing more to be done. If you get an equation 0 0, the system is dependent. You can then find the general solution by solving the first equation for x in terms of y..  The GaussJordan Method Matrices: We can replace a system of equations by an augmented matrix. Each row of the augmented matrix corresponds to an equation in the system. Each column, except for the last, corresponds to a variable.
5 The last column corresponds to the constants, and is usually separated from the others by a vertical line. To write a system of equations as a matrix: 1. Rearrange all of the equations so that all variables are on the left; all constants are on the right; and all variables are in the same order.. Then working from left to right, top to bottom, put the numbers from the equations into a matrix. If a variable is missing from a particular equation, put a zero as the corresponding entry of the matrix. The following three row operations may be performed on an augmented matrix without changing its solutions: 1. Interchange two rows.. Multiply all the entries in one row by a nonzero constant. 3. Replace one row by a nonzero multiple of itself plus a multiple of another row. The GaussJordan Method: To solve a system of equations, do the following: 1. Rearrange the equations into standard form.. Write the system as an augmented matrix. If there is a common factor in any row, divide it out. 3. In the first column, eliminate all entries except the first. ( Eliminate means Use row operations to make equal to zero.) If there is now a common factor in any row, divide it out. 4. In the second column, eliminate all entries except the second. Divide out common factors. 5. In the third column, eliminate all entries except the third. Divide out common factors. 6. Keep going until you ve dealt with every column. 7. Translate back into equations; the system should be solved. In the above procedure, it may be necessary to interchange two rows to get a nonzero entry in a diagonal position. If some row of your matrix becomes [ b] where b 0, the system is inconsistent. Game over. If you get a row consisting entirely of zeros, the system is dependent. To find the general solution, translate the matrix back into equations and solve in terms of a parameter. Word Problems:
6 First, you should check to see if the question is asking about a special situation which we ve studied in detail, eg. graphs, breakeven analysis or temperature conversion. In this case, apply the methods appropriate to that kind of question. Otherwise, here is a general strategy: 1. Identify the two or three (or very rarely, four) different quantities whose values you re being asked to find.. Assign a different letter to each of those quantities. Choose the letters so that it s obvious what they represent. 3. Use the information you re given to find some equations. You need to find as many equations as you have variables. 4. Solve these equations. 5. Interpret your solution back into words..3 Matrices  Addition and Subtraction Matrices Generalities: A matrix (not augmented) is just a rectangular array of numbers. Each number in the matrix is called an entry. Each horizontal line of entries is called a row. Each vertical line of entries is called a column. A matrix with m rows and n columns is called an m n matrix. For example, [ 1 ] is a 3 matrix. A matrix with just a single row is called a row matrix. A matrix with just a single column is called a column matrix. A matrix with just one row and just one column is more commonly known as a number. A matrix with the same number of rows as columns is called a square matrix. Two matrices are equal if they are of the same size, and each entry in one is equal to the corresponding entry in the other. Adding and Subtracting Matrices: You can only add or subtract matrices that have the same size. If two matrices have different sizes, their sum is undefined.
7 If A and B are two matrices of the same size, then their sum A + B is another matrix of the same size, each of whose entries is equal to the sum of the corresponding entries of A and B. Similarly, if A and B have the same size, then A B also has the same size as A and B, and the entries of A B are obtained by subtracting the entries of B from those of A. Basically, adding and subtracting matrices is defined in the obvious way. Just be careful with minus signs. The matrix whose entries are all 0 is called the zero matrix. There is a zero matrix of each size: eg the zero matrix, the 5 7 zero matrix, etc. The zero matrix behaves like the number 0, in the sense that A for every matrix A (the 0 in this equation denotes the zero matrix, not the number zero  you can t add a matrix and a number). If A is a matrix, then the matrix you get by negating (i.e. changing the sign in front of) each entry of A is called the negative of A, denoted A. This matrix has the property that A + ( A) 0, where 0 here means the zero matrix..4 Matrix Multiplication Scalar multiplication: Scalar means number. When you multiply a matrix by a number, you get a matrix of the same size as the one you started with. To multiply a matrix by a number, just multiply each entry of the matrix by that number. Be careful with negative signs. Row column: If R is a 1 n row matrix, and C is an n 1 column matrix, then RC is just a number (or in other words, a 1 1 matrix). To work out which number it is, multiply the first entry of R by the first entry of C; add the product of the second entries; add the product of the third entries; and so on. In symbols, c 1 [ ] c r1 r r 3... r n c 3 r 1 c 1 + r c + r 3 c r n c n.. c n
8 If the row and the column don t have the same number of entries, their product is undefined. The product CR, (i.e. in the opposite order to what we ve been talking about), is not the same as RC (in fact, the product CR will be an n n matrix, not a number). Matrix matrix: Warning: The product of two matrices is not defined in the obvious way: i.e. you don t just multiply corresponding entries together. If A is an m n matrix, and B is a p q matrix, then the product AB is only defined if n p; if this is the case, the product is an m q matrix. The way to remember this is: if you want to multiply A times B (in that order), write each matrix with its size underneath: A B m n p q The product AB is defined only if the inner numbers match (i.e. n p), and then the size of the product is given by the outside numbers (m q). Once you know the size of the product, calculate each entry using the formula: Entry in i th row, j th column of AB (i th row of A) (j th column of B), where the product on the right is a product row column as discussed above. Warning: Even if AB and BA are both defined, in most cases they will not be equal; they might even be of different sizes. So you need to be careful about the order in which you multiply. Matrix multiplication satisfies the following properties (assume in each case that the matrices are of the appropriate sizes, so that all of the products are defined) : 1. (AB)C A(BC). A(B + C) AB + AC 3. (A + B)C AC + BC 4. 0A A0 0, where 0 denotes the zero matrix. A square (n n) matrix which has 1 s down the main diagonal and zeros everywhere else, is called the n n identity matrix, denoted I n or just I. This matrix has the property that IA AI A for every n n matrix A. That is, in the world of n n matrices, the identity matrix I acts just like the number 1.
9 .5 Matrix Inverses Inverses Generalities: A square (n n) matrix is called invertible if there is another n n matrix B with the property that AB BA I n : i.e. when you multiply A by B on either side, you get the identity matrix. If such a matrix B exists, it is called the inverse of A, and is denoted by A 1. Not every matrix is invertible. For example, the zero matrix is never invertible; there are other (nonzero) matrices which are also not invertible. Determinants: The determinant of a square matrix A is a number, denoted by det(a). The matrix A is invertible if and only if det(a) does not equal zero. : For a matrix, the determinant is given by the formula ([ ]) a b det ad bc. c d 3 3: To find the determinant of a 3 3 matrix, copy the first two columns of the matrix to the right of the matrix. You then have three downward diagonals (each consisting of three numbers), and three upward diagonals (also consisting of three numbers each). Let d 1, d and d 3 denote the products of the downward diagonals; let u 1, u and u 3 denote the product of the upward diagonals. Then the determinant is given by det(a) (d 1 + d + d 3 ) (u 1 + u + u 3 ). Calculating inverses: Before you try to calculate the inverse of a matrix, check that it is invertible by calculating the determinant. : There is a formula swap the main diagonal, negate the offdiagonals and divide by the determinant: [ ] [ ] a b If A then A 1 1 d b c d det(a) c a 3 3: There is no formula (or there is, but it s horrible). To find the inverse of A, apply the GaussJordan method to the augmented matrix [A I 3 ]. When you ve got the lefthand side to look like I 3, the righthand side will be A 1. Check your answer by multiplying A 1 by A and making sure you get I 3.
10 Matrix Equations: Any system of linear equations can be written as a single matrix equation. The matrix equation will be of the form AX B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix for the system. The system has a unique solution if and only if A is invertible; in this case, the solution is given by X A 1 B. 3.1 Graphing linear inequalities Working with inequalities: An inequality can be rearranged just like an equation, except that whenever you multiply or divide both sides by a negative number, the inequality changes direction. Graphing a single linear inequality: 1. Draw the boundary line: replace the inequality by an, find the x and y intercepts, and then draw the line: If the inequality is strict (i.e. < or >), draw the boundary as a broken line this indicates that the boundary is not included in the region. If the inequality is nonstrict ( or ), use a solid line to indicate that the boundary is included in the region.. If the boundary is not vertical, decide if the feasible region is above or below the boundary. To do this, solve the inequality for y; If the result involves y or y >, then y is allowed to be large, so the feasible region is above the boundary. If the result is y or y <, then y is allowed to be small, so the region is below the boundary. 3. If the boundary is vertical (i.e. its equation is of the form x c), then decide whether the feasible region is to the left or to the right of the boundary, using the rule x > x is large feasible region is to the right. x < x is small feasible region is to the left. 4. Shade the appropriate region. Graphing a system of linear inequalities: 1. For each inequality, draw the boundary line and decide whether the feasible region is above or below (or left/right for a vertical boundary). Put some arrows on each boundary line to indicate which side the feasible region lies on.
11 . Once you ve drawn all the boundaries, the feasible region is the region where all the arrows point. Shade it. 3. Solving linear programming problems graphically Constraints and objectives: A linear programming problem involves: Variables: For us, there are always two variables, x and y. They represent two quantities whose values we can choose, and for which we seek the best choice. Constraints: Not all choices for the values of the variables will be possible; this is expressed by the constraints of the problem. The constraints are nothing but linear inequalities involving x and y. In any problem, there will usually be several constraints. Objective: The objective is what tells us which choices of x and y are better than others. In practice, there will be some quantity that we wish to maximise or minimise; this quantity will depend on (i.e. be a function of) x and y. We usually denote this function by the letter z; this is the objective function. In any problem, there will be only one objective function. How to solve linear programming problems: 1. Graph the feasible region determined by the constraints.. Find the coordinates of each corner of the feasible region. To do this, notice that each corner is the point of intersection of two boundary lines; solve the equations of those lines simultaneously to find the coordinates of the corner point. 3. The maximal and minimal values of the objective function occur at one of the corners. Draw a table with two columns, Corner and Value of the objective function. For each corner, plug the x and y coordinates in to the formula for the objective function to get the value of the function at that point. Enter the numbers into your table. 4. Look for the largest/smallest value of z (depending on whether the question asked you for a maximum or minimum, respectively). 5. Answer the question. For example, if you were asked to minimise z, and you found that the smallest value of z was 6 for the corner (5, 7), you would answer as follows: The minimal value of z is 6, which occurs at the point (5, 7).
12 3.3 Applications of linear programming (a.k.a. word problems) Linear programming can be used to solve realworld problems. These problems typically involve choosing the best combination of two quantities, subject to some constraints. To solve such a problem, 1. Read through the whole problem before you start writing anything.. Assign your variables: there will be two quantities whose values are to be chosen; call one of these x and the other y. 3. Determine the constraints: read through the question and look for any information that will give you an inequality. Key words to look out for are at least, at most, no more than, etc. Translate each such piece of information into an inequality involving x and y. Remember: In many cases, x and y will denote quantities of realworld objects. Such a quantity can never be negative; in such cases, you should add x 0 and y 0 to your list of constraints, even if this is not specifically mentioned in the problem. There will always be some information in the question that you won t use at this step; this comes up in the next step. 4. Find the objective function: The problem will always involve either maximising or minimising some quantity (eg. maximise revenue, minimise cost, etc.) Find an expression for this quantity in terms of x and y, and call it z. This is the objective function. 5. The problem has now been translated into algebra, and can be solved as in the previous section: graph the feasible region, find the corner points, make a table, find the maximum/minimum. 6. When you give your answer, you should translate back into words. For example, if x represents the number of product A produced, and y represents the number of product B produced, and the objective was the maximise profit, then you should give your final answer in the form The maximal profit is $15000, which can be attained by producing 300 of product A and 700 of product B, and not something like The maximal value of z is 15000, which occurs at the point (300, 700). 4 Linear transformations Note: This material is not covered in the textbook. In particular, it has nothing to do with Chapter 4 in your text. It will be covered on the exam. For more details, consult the notes you took in class. Basic notions and definitions: Recall from Section 1. that a linear function can be thought of as an object which takes a number x and transforms it into a new number, f(x).
13 A linear transformation is similar, but instead [ ] of transforming one number ([ into ]) another x x number, these transform one 1 matrix into another 1 matrix, T. y y Each linear transformation T is associated with a matrix A, in the sense that ([ ]) [ ] x x T A, y y where the righthand side is the usual product of a matrix with a 1 matrix. In other words, we transform one 1 matrix into another, by multiplying the original matrix by some given matrix A. ([ ]) [ ] x x The matrix T is called the image of under T. y y Examples:(1) Let A [ ] 1 image of under T. Solution: The image of [ ] 1 by multiplying by the matrix A: So the image of [ ] 3 4, and let T be the associated linear transformation. Find the 1 3 [ ] 1 under T means T T [ ] 1 under T is [ ] ([ ]) 1 [ ] [ ([ ]) 1. By definition, we compute this image ] [ ] 1 [ ] () Let A, and let T be the corresponding linear transformation. Find a matrix [ ] 0 whose image under T is. 9 Solution: We want to solve T ([ ]) x y [ ] 0, or in other words 9 [ ] [ ] 1 x 5 1 y [ ] 0. 9 Rewrite this matrix equation as a system of two linear equations: x y 0 and 5x y 9. Solve this system: subtract five times the first equation from the second, to get 9y 9. This implies that [ ] x y
14 y 1. Now plug this in to the first equation to get x 0, or in other words x. So the solution is [ ] [ ] x. y 1 You can check [ that ] this is correct by calculating the image of this point under T, and making sure 0 that it really is. 9 [ ] x Geometric description of linear transformations: We can think of a 1 matrix as corresponding to the point (x, y) in the plane. Consequently, we can think of a linear transformation as y a function which takes points in the plane and moves them to different points in the plane. There are several possibilities: Dilation: Every point (except the origin itself) is moved away from the origin in a straight line. Contraction: Every point (except the origin itself) is moved toward the origin along a straight line. Rotation: The points are rotated around the origin, but don t move toward or away from the origin. Reflection: All of the points on some straight line are fixed, while other points are moved to the opposite side of the line. A combination: Most linear transformations don t fall neatly into one of the above categories, but are a combination of the above. [ ] 0 1 Example: Let T be the linear transformation corresponding to the matrix. Which of the 1 0 following best describes T? (a) Dilation; (b) Contraction; (c) Rotation; (d) Reflection; (e) Some combination of the other options. [ ] [ ] [ ] [ ] [ ] Solution: Choose a few points, say,,,,, and calculate their images under T. For example, T ([ ]) 1 0 [ ] [ ] 1 0 so the image of is. Similarly, we get 0 1 ([ ]) [ ] 0 1 T, T 1 0 [ ] [ ] 1 0 [ ] 0, 1 ([ ]) [ ] ([ ]) 1 0 0, T [ ] 1, T 0 ([ ]) 1 1 [ ] 1. 1 Draw a picture which shows each point, along with its image under T and an arrow from each point to its image. You ll see that T moves each point to the point on the opposite side of the line y x, and leaves the points on this line where they are. So T is a reflection.
15 Linear transformations and lines: Fact: If T is a linear transformation, and L is a straight line, then the image of L under T is either a single point, or another straight line. (This is why they are called linear transformations.) To find the image of a given line: 1. Find two points on the original line: usually the x and yintercepts will do.. Calculate the images of these two points. 3. If these two image points are different, find the equation of the straight line which passes through those points. 4. If the two points you found in step are the same, then the image of the entire line is that same point. Example: Let T be the linear transformation associated to the matrix the line y x + under T. Solution: [ ] 1 0. Find the image of 1 1. Find two points on the original [ ] line: the yintercept is found by putting x [ 0; ] we get y 1, 0 so the line passes through. Similarly, we find that the xintercept is Calculate the images of these two points: we have ([ ]) [ ] [ ] T [ ] 0, ([ ]) [ ] [ ] [ ] 1 0 T [ ] 0 3. We know that the image of the given line passes through the points and equation of the line through these two points in the usual way: the slope is and we now use the pointslope formula, m y y 1 x x 1 0 4, y y 1 m(x x 1 ) y (x 0) y x So the image of the line y x + is the line y x +. [ ]. Find the Linear transformations and regions:
16 Fact: If T is a linear transformation and R is a region in the plane, then the image T(R) is either a region, a line segment, or a point. (The latter only happens when T is the zero matrix). To find the image of a region: 1. Find the image of each corner point.. Join these images in the same order as the original corners. 3. Shade the interior. Another fact: If T corresponds to the matrix A, then we have Area(T(R)) det(a) Area(R), where det(a) is the absolute value of the determinant of A: that is, it s just det(a) made positive  ignore the minus sign if there is one. Examples: (1) Let T be the linear transformation associated to the matrix A R be a region with area 3. Find the area of T(R). [ ] 3, and let 4 1 Solution: We have det(a) 3 ( 8) 5, and Area(R) 3, so by the usual formula, Area(T(R)) [ ] 3 0 () Let R be the region shown below. Use the linear transformation T with matrix A 1 1 to compute the area of R. Solution: Apply T to this region. The image is a rectangle with corners at the points [ ] [ ] 3 3, and. The sides of this rectangle have length 3 and, so we have Area(T(R)) 6. 0 The matrix A has determinant det(a) 3, so det(a) 3. Thus the formula gives us So the area of R is. Area(T(R)) det(a) Area(R) 6 3 Area(R) Area(R). [ ] 0, 0 [ ] 0,
Section 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (1,3), (3,3), (2,3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the xcomponent of a point in the form (x,y). Range refers to the set of possible values of the ycomponent of a point in
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationThe PointSlope Form
7. The PointSlope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationPRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71. Applications. F = mc + b.
PRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationEL9650/9600c/9450/9400 Handbook Vol. 1
Graphing Calculator EL9650/9600c/9450/9400 Handbook Vol. Algebra EL9650 EL9450 Contents. Linear Equations  Slope and Intercept of Linear Equations 2 Parallel and Perpendicular Lines 2. Quadratic Equations
More informationSolving a System of Equations
11 Solving a System of Equations 111 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined
More informationSection 1.4 Graphs of Linear Inequalities
Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationSituation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli
Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing
More informationApplied Finite Mathematics Second Edition. Rupinder Sekhon De Anza College Cupertino, California. Page 1
Applied Finite Mathematics Second Edition Rupinder Sekhon De Anza College Cupertino, California Page 1 Author: Rupinder Sekhon Associate Editors: Jessica and Vijay Sekhon Rupinder Sekhon has been teaching
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra  Linear Equations & Inequalities T37/H37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More informationThe slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6
Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means
More informationCHAPTER 1 Linear Equations
CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or xaxis, and the vertical axis or
More informationChapter 4.1 Parallel Lines and Planes
Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationIV. ALGEBRAIC CONCEPTS
IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other
More informationSlopeIntercept Equation. Example
1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the yintercept. Determine
More informationWeek 2 Quiz: Equations and Graphs, Functions, and Systems of Equations
Week Quiz: Equations and Graphs, Functions, and Systems of Equations SGPE Summer School 014 June 4, 014 Lines: Slopes and Intercepts Question 1: Find the slope, yintercept, and xintercept of the following
More informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationMath 113 Review for Exam I
Math 113 Review for Exam I Section 1.1 Cartesian Coordinate System, Slope, & Equation of a Line (1.) Rectangular or Cartesian Coordinate System You should be able to label the quadrants in the rectangular
More informationSolving Systems of Linear Equations. Substitution
Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationA synonym is a word that has the same or almost the same definition of
SlopeIntercept Form Determining the Rate of Change and yintercept Learning Goals In this lesson, you will: Graph lines using the slope and yintercept. Calculate the yintercept of a line when given
More informationBasic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.
Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required
More information1 Functions, Graphs and Limits
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xyplane), so this section should serve as a review of it and its
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationSome Lecture Notes and InClass Examples for PreCalculus:
Some Lecture Notes and InClass Examples for PreCalculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
More informationElements of a graph. Click on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and yintercept in the equation of a line Comparing lines on
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationUnit 1 Equations, Inequalities, Functions
Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1100 Overview: This unit models realworld situations by using one and twovariable linear equations. This unit will further expand upon pervious
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More information2.1 Equations of Lines
Section 2.1 Equations of Lines 1 2.1 Equations of Lines The SlopeIntercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is
More informationMath Review Large Print (18 point) Edition Chapter 2: Algebra
GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter : Algebra Copyright 010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,
More informationx x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =
Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More information1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.
1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points
More informationEdExcel Decision Mathematics 1
EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationLinear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationWe know a formula for and some properties of the determinant. Now we see how the determinant can be used.
Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationSolving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
More informationSect The SlopeIntercept Form
Concepts # and # Sect.  The SlopeIntercept Form SlopeIntercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not
More informationLinear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More informationChapter 1 Linear Equations and Graphs
Chapter 1 Linear Equations and Graphs Section 1.1  Linear Equations and Inequalities Objectives: The student will be able to solve linear equations. The student will be able to solve linear inequalities.
More informationP.E.R.T. Math Study Guide
A guide to help you prepare for the Math subtest of Florida s Postsecondary Education Readiness Test or P.E.R.T. P.E.R.T. Math Study Guide www.perttest.com PERT  A Math Study Guide 1. Linear Equations
More informationSection summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2
Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationGraphing Linear Equations in Two Variables
Math 123 Section 3.2  Graphing Linear Equations Using Intercepts  Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationSection 1.10 Lines. The Slope of a Line
Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes through
More informationHelpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:
Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants
More informationGraphing  Parallel and Perpendicular Lines
. Graphing  Parallel and Perpendicular Lines Objective: Identify the equation of a line given a parallel or perpendicular line. There is an interesting connection between the slope of lines that are parallel
More information2. Simplify. College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses
College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationChapter 2: Systems of Linear Equations and Matrices:
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
More informationThis assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the
More informationTexas Instruments TI83, TI83 Plus Graphics Calculator I.1 Systems of Linear Equations
Part I: Texas Instruments TI83, TI83 Plus Graphics Calculator I.1 Systems of Linear Equations I.1.1 Basics: Press the ON key to begin using your TI83 calculator. If you need to adjust the display contrast,
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationChapter 8. Quadratic Equations and Functions
Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property
More informationFlorida Algebra 1 EndofCourse Assessment Item Bank, Polk County School District
Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve
More informationInfinite Algebra 1 supports the teaching of the Common Core State Standards listed below.
Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More informationWeek 1: Functions and Equations
Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.12.2, and Chapter
More informationComputer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D Welcome everybody. We continue the discussion on 2D
More informationGeometry 1. Unit 3: Perpendicular and Parallel Lines
Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples
More informationPractice Problems for Exam 1 Math 140A, Summer 2014, July 2
Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Name: INSTRUCTIONS: These problems are for PRACTICE. For the practice exam, you may use your book, consult your classmates, and use any other
More informationSlopeIntercept Form of a Linear Equation Examples
SlopeIntercept Form of a Linear Equation Examples. In the figure at the right, AB passes through points A(0, b) and B(x, y). Notice that b is the yintercept of AB. Suppose you want to find an equation
More informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationLINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0
LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )
More information2.3 Writing Equations of Lines
. Writing Equations of Lines In this section ou will learn to use pointslope form to write an equation of a line use slopeintercept form to write an equation of a line graph linear equations using the
More informationThe Inverse of a Matrix
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationWriting the Equation of a Line in SlopeIntercept Form
Writing the Equation of a Line in SlopeIntercept Form SlopeIntercept Form y = mx + b Example 1: Give the equation of the line in slopeintercept form a. With yintercept (0, 2) and slope 9 b. Passing
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationLinear Programming Problems
Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number
More informationMatrix Algebra and Applications
Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2  Matrices and Matrix Algebra Reading 1 Chapters
More information