# Intersecting Two Lines, Part One

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Module 1.4 Page 97 of 938. Module 1.4: Intersecting Two Lines, Part One This module will explain to you several common methods used for intersecting two lines. By this, we mean finding the point x, y) at which two lines cross one another in the coordinate plane. Lines are actually graphical representations of linear equations in two variables often written in the form y = mx + b so what you will really be learning is how to solve systems of two linear equations by computing the x, y) values that make both equations simultaneously true. In the coordinate plane shown here, I have plotted two linear equations of the form y = mx + b, wherem and b are constants. The solid blue line maps the equation y 1 =4x + 1, while the dashed red line maps the equation y 2 = 4x + 9. If I were curious about where these two lines intersect on the plane, I could find the point at which they cross, which appears to lie directly on the gridline for x = 1, and exactly between the gridlines y = 4 and y = 6. I would speculate that the intersection between the two lines occurs at the solitary coordinate 1, 5). To test my possible solution, I plug x =1intothe linear equations, and hope to arrive at y = 5 for both. y 1 = 41) + 1 y 2 = 41) + 9 y 1 =4+1 y 2 = 4+9 y 1 =5 y 2 =5 yes!) yes!) This result tells me that at x = 1, both lines pass through the line y = 5. Therefore, the two lines intersect one another at and only at the coordinate 1, 5). Examining a graph, as we just did, is certainly useful when one wants to approximate the intersection of two lines. However, this method is not reliable for finding exact solutions which is what we desire) when the linear equations intersect at decimal-valued coordinates. For example, the lines y 1 =0.6x 3 and y 2 =3x intersect at the x, y) coordinate 6, 3.7). This is not a point that one can derive by simply eyeballing the graph, especially when gridlines are sparse or nonexistent. And guessing-and-checking various approximations to the true x, y) solution is neither an e cient nor useful expenditure of your time, so it will be strictly avoided. Rather, this module will cover purely algebraic methods for intersecting two lines, which guarantee exact solutions every time if solutions exist). Nonetheless, an occasional graph can help with visualizing a particular situation. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

2 Module 1.4 Page 98 of 938. A Pause for Reflection... If I include a graph in the remainder of this module, it is only to illustrate some mathematical theory related to the behavior between lines on a plane. Any solutions we obtain throughout this module will be the result of computation with reliable algebraic methods. We will utilize the unbiased rules of algebra, and not rely on our fallible eyesight, so we can be certain that our solutions to the following examples will be 100% accurate the first time around. A single line on the plane is a graphical representation of the linear equation y = mx + b, wherem is the slope rate of change) of the line and b is the y-value at which the line intersects the y-axis. The line graphed to the left is y =2.5x + 1. Any solution to a two-variable linear equation is an x, y) coordinate pair that, when plugged into the original equation, yields a true mathematical expression. For example, an obvious solution point is the coordinate 0, 1): if you plug x = 0 and y = 1 into the equation, you obtain the true statement 1 = 1. Can you tell whether a coordinate pair lies on a particular line? For each bullet, state whether the given x, y) pair is a solution to the corresponding linear equation. Note the various ways to express your answer. Is 3 2, 8.5) a solution to y =7x 2? [Answer: Yes, that pair is a solution to the linear equation.] Does 45, 750) lie on y = 20x + 180? [Answer: No, that point does not lie on the line.] # Does the line y =5x + 20 contain the point 4, 0)? [Answer: Yes, the line contains that point.] Is the coordinate 4, 8 3 ) on the line y = 5 3 x 9? [Answer: No, that coordinate is outside the line.] It is misleading, however, to ask for a solution to a single linear equation in two variables. A line is composed of an infinite number of points, thus there are an infinite number of solutions. Furthermore, the equation for the line is itself a compact expression of the complete collection of solutions for any x-value you plug in, you can find its corresponding y-value on the line, and thus have a solution point. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

3 Module 1.4 Page 99 of 938. To illustrate the above Danger! box, I have plotted the line y =2.5x + 1 and highlighted a number of individual points that lie on the line. Any of these coordinate points can be plugged into the linear equation to produce a mathematical truth. In fact, every point on this line of which there are infinitely many is a solution to the linear equation. This is the case for any single equation of the form y = mx + b. A more interesting endeavor is discovering the solution to a system of linear equations. A system of linear equations is a collection of two or more linear equations which you must simultaneously solve. It can be represented as two or more lines on the coordinate plane. Except for two rare situations which we will discuss very shortly), a pair of lines will intersect at exactly one coordinate point. Plugging this pair of x, y)- values into either of the individual equations will yield a true mathematical expression. The graph on the left plots the linear system y = x +1 y = 2x +6 The lines intersect or, the system is solved at the x, y) coordinate 5 3, 8 3 ). We will check this solution in the next box. In a couple of pages, you will learn how to determine the solution to a linear system yourself. You should check that the point 5 3, 8 3 ) does in fact represent the solution to the above linear system. To do this, plug the x- and y-values into both equations and see that a true expression is produced from both. y = x +1 y = 2x = = = = 8 3 yes!) 8 3 = = 8 3 yes!) As you can see, plugging the coordinate point into each original equation produced the true, trivial statement 8 3 = 8 3. This confirms that the point 5 3, 8 3 ) is the intersection point of the two lines. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

4 Module 1.4 Page 100 of 938. Assume you are given a system of linear equations and a coordinate pair x, y). To determine whether the coordinate is a solution to the system, you must plug the x- and y- values into both linear equations and check that a true mathematical expression follows from each. This will confirm that the two lines intersect at that coordinate. If only one, or neither, of the equations produces a true expression from the variable substitution, then the coordinate pair x, y) is not a solution to the system. Consider the linear system y = x + 10 y =1.5x 10 Is the coordinate pair 8, 2) a solution to the system? potential solution into both equations of the system. To find out, you must plug the y = x + 10 y =1.5x 10 2) = 8) ) = 1.58) 10 2= = =2 2=2 yes!) yes!) Both resulting expressions are true, which assures us that the point 8, 2) is a solution. The fact that both true expressions are 2 = 2 is a consequence of the linear equations being identically set up in the form y = mx + b. For each bullet, determine whether the given coordinate pair solves the linear system. y =3x 2 Does 1, 5) solve the system? y = x 6 [Answer: Yes, this is the solution to the system.] y = 2x +9 Does 4, 1) solve the system? y =3x 16 [Answer: No, this is not the solution to the system.] # Note, we really did have to check both equations in the second bullet, because y = 2x + 9 was satisfied while y =3x 16 was not satisfied. Since both equations must be satisfied, we reject 4, 1) as a solution to the system. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

5 Module 1.4 Page 101 of 938. Four boxes ago I presented the system of linear equations y = x +1 y = 2x +6 It is in our interest to allow the equations in the system to interact with one another. Since we are looking for only one x, y)-pair that solves the system, then the y-values must be equal. Using this knowledge, we can rewrite the system as x + 1 = y = 2x + 6. Now, since x + 1 equals y and 2x + 6 also equals y, then it must be the case that x + 1 equals 2x + 6. Thus, we can write the equation x+1 = 2x+6. We solve this one-variable linear equation to obtain the x-value of our solution to the original system. We will proceed with solving this linear system in the next box. We now have a linear equation in one variable, x +1= as follows: 2x + 6. We can solve that easily, x +1 = 2x +6 x +2x = 6 1 3x = 5 x = 5 3 We obtain the non-trivial expression x = 5 3, which tells us that the two lines from the system of equations intersect at x = 5 3. Next, to find the y-value of the point of intersection, we plug x = 5 3 into one of the equations from the original system. y = y = 8 3 The substitution of x = 5 3 yielded y = 8 3. In other words, the two lines y = x + 1 and 5 y = 2x + 6 intersect at the point 3, 8 3. Checking the solution to a linear system is a matter of determining whether the lines share the same y-value at a particular x-value. In the previous box, we plugged our known x-value into one of the equations to find the corresponding y-value. To check your work, it is best practice to then plug the x-value into the other equation in the system, to see whether the same y-value is produced. If the resulting y-value is a match, then congratulations! you have found the complete solution to your system. 5 Let s check that 3, 8 3 is a solution to the system by plugging x = 5 3 into y = 2x + 6: y = y = y = 8 3 Since we also obtained y = 8 3 by plugging into y = x+1, we know that the two lines intersect at x = 5 3. Our solution is now confirmed. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

6 Module 1.4 Page 102 of 938. In general, when solving a system of linear equations where one variable is known, you need only plug that value into one of the two original equations to find the other variable. Then you can check your work by plugging the solution point into the unused equation to see whether a true expression results. Let s try solving another system of linear equations. This time our system is y =4x 9 y =6 x If a solution exists, the y s in both equations should be equal. Thus, we can collapse the system into a linear equation in x only and solve for x: 4x 9=6 x 4x + x =6+9 5x = 15 x =3 Having obtained x = 3, we then plug this value back into one of the two original equations to determine the corresponding y-value. Let s arbitrarily choose the first equation: y = 43) 9 y = 12 9 y =3 Now we have a potential solution point for our system, 3, 3). To confirm that the coordinate 3, 3) describes the intersection of the two lines in the above example, we plug the solution into the other, untested linear equation to check that a true mathematical expression is produced: y =6 x 3) = 6 3) 3=3 yes!) The validity of the resulting expression 3 = 3 confirms that the second line contains the point 3, 3), which we found to also be contained within the first line. Therefore, the two lines do intersect at 3, 3). Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

7 Module 1.4 Page 103 of 938. # Find the point where the following two lines intersect: y =0.2x 24 y = 0.5x + 11 [Answer: The two lines intersect at the point 50, 14).] # Find the point where the following two lines intersect: y = 41x + 58 y = 17x + 29 [Answer: The two lines intersect at the point 0.5, 37.5).] # Find the solution to the linear system: y = 33x 4 y = 21x 10 [Answer: The approximate solution to the linear system is x = 0.111,y = ] # Find the solution to the system of linear equations: y =5x 30 y = 3x + 23 [Answer: The system is solved when x =6.625 and y =3.125.] When we are given a system of linear equations, we first assume that a solution exists for the system. That is, we expect the lines expressed by the linear equations to intersect somewhere in the coordinate plane. Working from that assumption, we first try to find the value of one of the variables so far we have found x first, but that need not be the case, as you will discover shortly). If a solution exists for the first variable, this confirms our assumption that the lines intersect. What we are about to show is that if the variable disappears while solving, we must concede that the lines either do not intersect or are not unique; in either of these peculiar scenarios, there is no single solution to the system. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

8 Module 1.4 Page 104 of 938. So far, you have only seen systems of linear equations for which a single solution exists. However, I have mentioned the caveat that a solution may not exist for some systems. In fact, there is a third scenario that there are an infinite number of solutions to a system of linear equations. We will now explore all three scenarios in detail. Scenario One Common): Single Solution A system of two linear equations in standard form is y 1 = m 1 x 1 + b 1 y 2 = m 2 x 2 + b 2 where m is the slope and b is the y-intercept of a line. If, for a system in standard form, m 1 6= m 2 the slopes are di erent), the system will have one unique solution. The graphical representation of the system will be two lines crossing one another at exactly one point on the plane. The graph on the left plots the system of equations y =2x 3 y = 2x +1 where it is clear that the lines have di erent slopes. Thus we would expect an intersection at only one point in the plane. When a system of linear equations has exactly one solution, it is called an independent system. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

9 Module 1.4 Page 105 of 938. Scenario Two Rare): No Solutions See the model of a linear system in standard form, shown above. When m 1 = m 2 but b 1 6= b 2, the system has no points of intersection no solutions). This is because the lines are parallel but disjoint, running along forever in both directions without ever crossing. Think of it this way: the lines have the same slope, but they cross the y-axis at di erent heights. Thus as you shift horizontally over the coordinate plane, both lines increase or decrease at the same rate; they are always separated vertically by the di erence between b 1 and b 2. The graph on the left plots the system of equations y =2x 3 y =2x 1 We can see that the slopes of the lines match, but the y- intercepts do not. To be hyper-specific, the y-coordinate of any point on the blue solid line, for any particular value of x, will always be two less than the y-coordinate of the corresponding point on the red dashed line, for that same value of x. Therefore, this system has no solution. When a system of linear equations has no solution, it is called an inconsistent system. If a system of linear equations is presented to you in standard form, it should be very apparent whether the lines are parallel or intersecting; simply compare the lines slopes. However, there is another tell-tale sign that a system has no solutions: when solving for the first unknown variable, a nonsensical expression results rather than a true algebraic assignment. We will see an example of this in the next box. Consider the system of linear equations y =7x 5 y =7x +2 Since the slopes of the lines are equal but their y-intercepts are not, we can automatically deduce that the lines are parallel and thus do not intersect. However, let s collapse this system into a one-variable linear equation and see what we get when we try to solve it. 7x 7x 5 7x? = 7x +2? = = 7 Our result 0 = 7 is certainly false. And where did the variable x go? These errors arose for one reason only: we were trying to solve for an inconsistent system. We can conclude that the system has no solution. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

10 Module 1.4 Page 106 of 938. # Determine the number of solutions for each linear system. Then classify each system. y =4x 3 [Answer: The system has no solutions. It is an inconsistent system.] y =2+4x y = 5 x [Answer: The system has one solution. It is an independent system.] y = x 10 If I asked you to compare the two linear equations y =5x+4 and y =5x+4, you d probably give me a sideways glance and tell me that they are the same line, obviously. But what about y =5x + 4 and 4y = 20x + 16? Does it surprise you that these also describe the same exact line that if you graphed these two linear equations on the coordinate plane, they would overlap perfectly? Linear equations are, first and foremost, equations. And equations can be manipulated, including being scaled multiplied) by a non-zero constant term. Notice that 4y = 20x + 16 is just y =5x+4 with each of its terms multiplied by 4. Although the coe cients are larger, the linear relationship between y and x is preserved, which is what matters. In short, if you can scale a linear equation multiplying through by a constant term so that it matches another linear equation, then the two equations are actually equivalent as objects lines) on the coordinate plane. Scenario Three Rare): Infinitely Many Solutions If, in a system of linear equations, one equation is a constant multiple of the other, then the two equations actually describe the same line. When graphed on the coordinate plane, the two lines will overlap one another. That the lines overlap everywhere on the plane is another way of saying they intersect at all points. Therefore, a system of linear equations in which one equation is a constant multiple of the other has an infinite number of solutions: every point on either line is a solution to the system. The graph to the left plots the system of equations y =2x 3 3y =6x 9 We can see that the second equation is a multiple of the first by a factor of 3), and that their corresponding lines overlap completely. Every point on the line y =2x 3is a solution to the the system. When a system of linear equations has an infinite number of solutions, it is called a dependent system. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

11 Module 1.4 Page 107 of 938. Find the point of intersection between the two lines described by the following equation: 2.5y =5x y =3x 7.5 First, I will get both equations into standard form, so that I can compare them. The top equation can be divided by 2.5, and the bottom equation can be divided by 1.5. I should reiterate that scaling an equation either larger or smaller does not alter the relationship between y and x in that equation. After scaling, the system of equations is rewritten as y =2x 5 y =2x 5 However, we can now see that these are the same line. Therefore, any point on the line y =2x 5 is a solution to the system. The system has an infinite number of solutions, represented by the linear equation y = 2x 5. You might now be tired of solving systems of two linear equations. However, we will see many word problems that are solved by these types of equations in Sections..., so it is beneficial to practice this skill just a little more. Solve the following system of linear equations, and then describe the system using one of the three scenarios introduced above: y = x y = 3.3x +3.3 # [Answer: The system has no solutions, and therefore is an inconsistent system.] Solve the following system of linear equations, and then describe the system using one of the three scenarios introduced above: y =1.3x + 30 y = 1 2 x + 18 # [Answer: The solution to this system is x = 6.66, y = This system is independent.] Solve the following system of linear equations, and then describe the system using one of the three scenarios introduced above: y = 100x + 25 y = x # [Answer: This system has an infinite number of solutions, and is therefore dependent.] Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

12 Module 1.4 Page 108 of 938. So far you have only seen linear systems in which both equations were already in standard form. When this is the case, solving is only a matter of setting the polynomials in x equal to one another, solving for x, thenusingx to solve for y. What if a system of equations is not presented in standard form? What if you were asked, for example, to find the intersection of 10 = 3x 9y and 2x =5+ 4 5y? Luckily, there are two very popular algebraic methods available to solve systems where the equations are not in standard form; they are called the Substitution Method and the Elimination Method. The next module will focus on teaching you these methods, so that you can most e ciently solve any system of linear equations, no matter how complexly they are setup. In this module we reviewed: how a system of linear equations in two variables represents a set of lines in the coordinate plane. the notion of a solution to a system of linear equations, and how it represents the point at which the lines cross in the plane. a simple method for solving systems of linear equations or finding where two lines intersect when the equations are in standard form. the characteristics of independent, inconsistent, and dependent linear systems. the idiosyncrasies that arise when trying to find solutions for an inconsistent or dependent system. the following vocabulary terms: system of linear equations, independent system, inconsistent system, dependent system. Commons License specifically agreement # 3 attribution and non-commercial. ) Until such time as the document is completed, however, the

### Section 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,,

### Section 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a.

Algebra 1 Chapter 7 Notes Name Section 7.1 Solving Linear Systems by Graphing System of Linear Equations: Two or more equations in the same variables, also called a. Solution of a System of Linear Equations:

### EQUATIONS 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

### 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 y = mx + b.

PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 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

### Solving Systems of Two Equations Algebraically

8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns

### 5.1. Systems of Linear Equations. Linear Systems Substitution Method Elimination Method Special Systems

5.1 Systems of Linear Equations Linear Systems Substitution Method Elimination Method Special Systems 5.1-1 Linear Systems The possible graphs of a linear system in two unknowns are as follows. 1. The

### Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

### Lesson 22: Solution Sets to Simultaneous Equations

Student Outcomes Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically. Classwork Opening Exercise

### 2x - y 4 y -3x - 6 y < 2x 5x - 3y > 7

DETAILED SOLUTIONS AND CONCEPTS GRAPHICAL REPRESENTATION OF LINEAR INEQUALITIES IN TWO VARIABLES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu.

### In this section, we ll review plotting points, slope of a line and different forms of an equation of a line.

Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane:

### Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### Graphing 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

### Solving Equations Involving Parallel and Perpendicular Lines Examples

Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines

### Lecture 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

### Linear 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 x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

### Chapter 2. Linear Equations

Chapter 2. Linear Equations We ll start our study of linear algebra with linear equations. Lots of parts of mathematics arose first out of trying to understand the solutions of different types of equations.

### x 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

### 3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes

Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### 2. THE x-y PLANE 7 C7

2. THE x-y 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

### graphs, Equations, and inequalities

graphs, Equations, and inequalities You might think that New York or Los Angeles or Chicago has the busiest airport in the U.S., but actually it s Hartsfield-Jackson Airport in Atlanta, Georgia. In 010,

### 1. Graphing Linear Inequalities

Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means

### The Cartesian Plane The Cartesian Plane. Performance Criteria 3. Pre-Test 5. Coordinates 7. Graphs of linear functions 9. The gradient of a line 13

6 The Cartesian Plane The Cartesian Plane Performance Criteria 3 Pre-Test 5 Coordinates 7 Graphs of linear functions 9 The gradient of a line 13 Linear equations 19 Empirical Data 24 Lines of best fit

### Study Guide and Review - Chapter 4

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The y-intercept is the y-coordinate of the point where the graph crosses the y-axis. The

### Slope-Intercept 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 y-intercept. Determine

### COMPARING LINEAR AND NONLINEAR FUNCTIONS

1 COMPARING LINEAR AND NONLINEAR FUNCTIONS LEARNING MAP INFORMATION STANDARDS 8.F.2 Compare two s, each in a way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example,

### CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS

CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS Specific Expectations Addressed in the Chapter Collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology

### Pre-AP Algebra 2 Lesson 2-5 Graphing linear inequalities & systems of inequalities

Lesson 2-5 Graphing linear inequalities & systems of inequalities Objectives: The students will be able to - graph linear functions in slope-intercept and standard form, as well as vertical and horizontal

### Vocabulary 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

### Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - 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.

### 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

### Study Resources For Algebra I. Unit 1D Systems of Equations and Inequalities

Study Resources For Algebra I Unit 1D Systems of Equations and Inequalities This unit explores systems of linear functions and the various methods used to determine the solution for the system. Information

### Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 173.

Algebra Chapter : Systems of Equations and Inequalities - Systems of Equations Chapter : Systems of Equations and Inequalities System of Linear Equations: - two or more linear equations on the same coordinate

### A synonym is a word that has the same or almost the same definition of

Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

### 7.5 Systems of Linear Inequalities

7.5 Systems of Linear Inequalities In chapter 6 we discussed linear equations and then saw linear inequalities, so since here, in chapter 7, we are talking about systems of linear equations, it makes sense

### Section 3.4 The Slope Intercept Form: y = mx + b

Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept Reminding! m = y x = y 2 y 1 x 2 x 1 Slope of a horizontal line is 0 Slope of a vertical line is Undefined Graph a linear

### Section 3.2. Graphing linear equations

Section 3.2 Graphing linear equations Learning objectives Graph a linear equation by finding and plotting ordered pair solutions Graph a linear equation and use the equation to make predictions Vocabulary:

### Write the Equation of the Line Review

Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections

### Helpsheet. Giblin Eunson Library LINEAR EQUATIONS. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet Giblin Eunson Library LINEAR EQUATIONS Use this sheet to help you: Solve linear equations containing one unknown Recognize a linear function, and identify its slope and intercept parameters Recognize

### Patterns, Equations, and Graphs. Section 1-9

Patterns, Equations, and Graphs Section 1-9 Goals Goal To use tables, equations, and graphs to describe relationships. Vocabulary Solution of an equation Inductive reasoning Review: Graphing in the Coordinate

### Standard Form of Quadratic Functions

Math Objectives Students will be able to predict how a specific change in the value of a will affect the shape of the graph of the quadratic ax bxc. Students will be able to predict how a specific change

### Systems of Linear Equations

DETAILED SOLUTIONS AND CONCEPTS - SYSTEMS OF LINEAR EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE

### Equations of Lines Derivations

Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated

### TEKS 2A.7.A Quadratic and square root functions: connect between the y = ax 2 + bx + c and the y = a (x - h) 2 + k symbolic representations.

Objectives Define, identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems. Vocabulary axis of symmetry standard form minimum value maximum

### Sect The Slope-Intercept Form

Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept 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

### 8.1 Systems of Linear Equations: Gaussian Elimination

Chapter 8 Systems of Equations 8. Systems of Linear Equations: Gaussian Elimination Up until now, when we concerned ourselves with solving different types of equations there was only one equation to solve

### 2.1 Systems of Linear Equations

. Systems of Linear Equations Question : What is a system of linear equations? Question : Where do systems of equations come from? In Chapter, we looked at several applications of linear functions. One

### Creating Equations. Set 3: Writing Linear Equations Instruction. Student Activities Overview and Answer Key

Creating Equations Instruction Goal: To provide opportunities for students to develop concepts and skills related to writing linear equations in slope-intercept and standard form given two points and a

### LINEAR EQUATIONS IN TWO VARIABLES

66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

### 0-5 Systems of Linear Equations and Inequalities

21. Solve each system of equations. Eliminate one variable in two pairs of the system. Add the first equation and second equations to eliminate x. Multiply the first equation by 3 and add it to the second

### Systems of Linear Equations in Two Variables

Chapter 6 Systems of Linear Equations in Two Variables Up to this point, when solving equations, we have always solved one equation for an answer. However, in the previous chapter, we saw that the equation

### 5.2. Systems of linear equations and their solution sets

5.2. Systems of linear equations and their solution sets Solution sets of systems of equations as intersections of sets Any collection of two or more equations is called a system of equations. The solution

### Math 181 Spring 2007 HW 1 Corrected

Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the x-axis (horizontal axis)

### GRAPHING LINEAR EQUATIONS IN TWO VARIABLES

GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: Slope-Intercept Form: y = mx+ b In an equation

### Introduction to Diophantine Equations

Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field

### 1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient

Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear

### Lesson Plan Mine Shaft Grade 8 Slope

CCSSM: Grade 8 DOMAIN: Expressions and Equations Cluster: Understand the connections between proportional relationships, lines, and linear equations. Standard: 8.EE.5: Graph proportional relationships,

### 5 Systems of Equations

Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

### Equations 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.

### GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

### Chapter 5.1 Systems of linear inequalities in two variables.

Chapter 5. Systems of linear inequalities in two variables. In this section, we will learn how to graph linear inequalities in two variables and then apply this procedure to practical application problems.

### Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called.

Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate

### Topic 15 Solving System of Linear Equations

Topic 15 Solving System of Linear Equations Introduction: A Linear Equation is an equation that contains two variables, typically x and y that when they are graphed on a cartesian grid make a straight

### Linear Equations Review

Linear Equations Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The y-intercept of the line y = 4x 7 is a. 7 c. 4 b. 4 d. 7 2. What is the y-intercept

### 2. SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the y-values.

CCSS STRUCTURE State the domain and range of each relation. Then determine whether each relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. 2. The members

### Student Lesson: Absolute Value Functions

TEKS: a(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete,

### Section 2.1 Intercepts; Symmetry; Graphing Key Equations

Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the x-axis. 2. The x-coordinate of a point

### A correlation exists between two variables when one of them is related to the other in some way.

Lecture #10 Chapter 10 Correlation and Regression The main focus of this chapter is to form inferences based on sample data that come in pairs. Given such paired sample data, we want to determine whether

### with "a", "b" and "c" representing real numbers, and "a" is not equal to zero.

3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,

### Unit 1 Equations, Inequalities, Functions

Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1-100 Overview: This unit models real-world situations by using one- and two-variable linear equations. This unit will further expand upon pervious

### Use the graph shown to determine whether each system is consistent or inconsistent and if it is independent or dependent.

Use the graph shown to determine whether each system is consistent or inconsistent and if it is independent or dependent. y = 3x + 1 y = 3x + 1 The two equations intersect at exactly one point, so they

### Linear Equations and Graphs

2.1-2.4 Linear Equations and Graphs Coordinate Plane Quadrants - The x-axis and y-axis form 4 "areas" known as quadrants. 1. I - The first quadrant has positive x and positive y points. 2. II - The second

### Elements 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 y-intercept in the equation of a line Comparing lines on

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

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### Building Polynomial Functions

Building Polynomial Functions NAME 1. What is the equation of the linear function shown to the right? 2. How did you find it? 3. The slope y-intercept form of a linear function is y = mx + b. If you ve

### Session 7 Bivariate Data and Analysis

Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares

### Answer on Question #48173 Math Algebra

Answer on Question #48173 Math Algebra On graph paper, draw the axes, and the lines y = 12 and x = 6. The rectangle bounded by the axes and these two lines is a pool table with pockets in the four corners.

### Positive numbers move to the right or up relative to the origin. Negative numbers move to the left or down relative to the origin.

1. Introduction To describe position we need a fixed reference (start) point and a way to measure direction and distance. In Mathematics we use Cartesian coordinates, named after the Mathematician and

### SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

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

### Writing the Equation of a Line in Slope-Intercept Form

Writing the Equation of a Line in Slope-Intercept Form Slope-Intercept Form y = mx + b Example 1: Give the equation of the line in slope-intercept form a. With y-intercept (0, 2) and slope -9 b. Passing

### Temperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.

Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is

### The 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

### Lines and Linear Equations. Slopes

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

### The Graph of a Linear Equation

4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

### Functions and straight line graphs. Jackie Nicholas

Mathematics Learning Centre Functions and straight line graphs Jackie Nicholas c 004 University of Sydney Mathematics Learning Centre, University of Sydney 1 Functions and Straight Line Graphs Functions

### Let s explore the content and skills assessed by Heart of Algebra questions.

Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,

### FUNCTIONS. Introduction to Functions. Overview of Objectives, students should be able to:

FUNCTIONS Introduction to Functions Overview of Objectives, students should be able to: 1. Find the domain and range of a relation 2. Determine whether a relation is a function 3. Evaluate a function 4.

1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

### Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

### Answer Key Building Polynomial Functions

Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,

### Additional Examples of using the Elimination Method to Solve Systems of Equations

Additional Examples of using the Elimination Method to Solve Systems of Equations. Adjusting Coecients and Avoiding Fractions To use one equation to eliminate a variable, you multiply both sides of that

### Many different kinds of animals can change their form to help them avoid or

Slopes, Forms, Graphs, and Intercepts Connecting the Standard Form with the Slope-Intercept Form of Linear Functions Learning Goals In this lesson, you will: Graph linear functions in standard form. Transform

### Situation 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

Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x

### CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided

### Activity 6. Inequalities, They Are Not Just Linear Anymore! Objectives. Introduction. Problem. Exploration

Objectives Graph quadratic inequalities Graph linear-quadratic and quadratic systems of inequalities Activity 6 Introduction What do satellite dishes, car headlights, and camera lenses have in common?