Anytime plan TalkMore plan
|
|
- Emmeline Harrison
- 7 years ago
- Views:
Transcription
1 CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations is a set of two or more equations with the same variables. A solution of a sstem of equations is a set of values that makes all the equations true. Read the eample in our book and then read the eample below. EXAMPLE The Antime long-distance plan charges $.8 per month plus 5 a minute. The TalkMore plan charges 9 a minute and no monthl fee. For what number of minutes are the charges for the two plans the same? a. Write a sstem of two equations to model this situation. b. Solve the sstem b creating a table. Eplain the real-world meaning of the solution, and locate the solution on a graph. Solution a. Let represent the number of minutes, and let represent the charge in dollars. The charge is the monthl fee plus the rate times the number of minutes. Here is the sstem of equations. b. Create a table from the equations. Fill in the times and calculate the charge for each plan. The table shows that when, both -values are.8. Because (,.8) satisfies both equations, it is the solution of the sstem. The solution means that both plans charge $.8 for minutes of long-distance calls. On the graph, the solution is the point where the two lines intersect. Charge (dollars) Antime (,.8) TalkMore (, ) Antime plan TalkMore plan (, 18) 8 16 Time (minutes) (, 1.8) Long-Distance Plans Time Antime TalkMore (min) (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 75
2 Previous Net Lesson 6.1 Solving Sstems of Equations (continued) Investigation: Where Will The Meet? Steps 1 In this investigation, two students walk along a 6-meter segment. Walker A starts at the.5-meter mark and walks toward the 6-meter mark at a rate of 1 m/sec. Walker B starts at the -meter mark and walks toward the 6-meter mark at a rate of.5 m/sec. Here is a graph of the data collected b one group. Steps 5 7 You can model this situation with a sstem of equations and then solve the sstem to figure out when and where Walker A passes Walker B. If represents the time in seconds and represents the distance from the -meter mark, the sstem is.5.5 Here are graphs of the equations on the same aes. The graphs intersect at (,.5), indicating that Walker A passes Walker B after seconds, when both walkers are at the.5-meter mark. Steps 8 If Walker A moved faster than 1 m/sec, the slope of Walker A s line would increase and the intersection point would move closer to the origin, indicating that Walker A passes Walker B sooner and closer to the -meter mark. If the two walkers moved at the same speed, the would never meet. The slope of the lines would be equal, so the lines would be parallel. The sstem of equations for this situation has no solution. 1 Walker A Walker B 1 Distance (m) Distance (m) A A A B B A B A B B A Time (sec) Walker A Walker B Time (sec) Distance (m) 8 6 Walker A Walker B Distance (m) 8 6 Walker B Walker A 6 8 Time (sec) Time (sec) 1 If both walkers walked at the same speed from the same starting mark, the two lines would be identical. Ever point on the line is a solution of the sstem, indicating that the walkers are alwas at the same location at the same time. The investigation shows that two lines can intersect at zero points, at one point, or at ever point. So a sstem of linear equations can have zero, one, or an infinite number of solutions. 76 Discovering Algebra Condensed Lessons Ke Curriculum Press
3 CONDENSED L E S S O N 6. Solving Sstems of Equations Using Substitution Previous Net In this lesson ou will represent situations with sstems of equations use the substitution method to solve sstems of linear equations When ou use a graph or a table to solve a sstem of equations, ou ma onl be able to find an approimate solution. The substitution method allows ou to find an eact solution of a sstem. Read Eample A in our book, which shows how to solve a sstem using the substitution method. Investigation: All Tied Up Start with a thin rope and a thick rope, each 1 meter long. If ou tie knots in each rope, measuring the length after each knot, ou might get data like this. Use the techniques ou learned in Chapter 5 to write a linear equation to model the data for each rope. A possible model for the thin rope is 6, where is the number of knots and is the length in centimeters. The -intercept,, is the length of the rope before ou tie an knots. The slope, 6, is the change in the length for each knot. A possible model for the thick rope is.. This equation indicates that the initial length is cm and that the length decreases b. cm for each knot. Now, suppose the initial length of the thin rope is 9 meters and the initial length of the thick rope is meters. This sstem of equations models this situation To estimate the solution of this sstem, make a graph and estimate the point of intersection. The intersection point is about (, 76). You can also find the solution b using the substitution method. Substitute 9 6 (from the first equation) for in the second equation, and solve the resulting equation.. Length of thin rope Length of thick rope Original second equation Substitute 9 6 for. Number of knots 9. Add 6 to both sides and simplif. Thin Rope Length (cm) Length of ropes (cm) 8 6 Number of knots Thick Rope Length (cm) m thin rope -m thick rope Number of knots. Subtract from both sides..6 Divide both sides b.. (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 77
4 Previous Net Lesson 6. Solving Sstems of Equations Using Substitution (continued) Because represents the number of knots, the solution must be a whole number. So round to. When is, is about 76. So the solution is (, 76). This means that when knots have been tied in each rope, the ropes are about the same length, 76 cm. Think about how the models would be different if the two ropes had the same thickness. In this situation the slopes would be the same, so the lines would be parallel. In this case the sstem would have no solutions. In other words, the ropes would never be the same length. If the ropes had the same thickness and the same starting length, the equations and the lines would be eactl the same. In this case there are man solutions. The ropes would be the same length after an number of knots had been tied. When ou solve a sstem using the substitution method, ou sometimes need to rewrite one of the equations before ou can substitute. Eample B in our book shows ou how to solve a sstem when both equations are given in standard form. Read this eample and the tet that follows carefull. Then, read the eample below. EXAMPLE Use the substitution method to solve this sstem. s 5t s 6 t 5 Solution Rewrite one of the equations so that a variable is alone on one side. s 6 t 5 s 11 t Original second equation. Subtract 5 from both sides. Now, substitute s 11 for t in the first equation and solve for s. s 5(s 11) Substitute s 11 for t. s s 55 Distribute the 5. s s 5 Subtract from 55. 1s 5 Add s to both sides. s Divide both sides b 1. To find the value of t, substitute for s in either equation and solve for t. You should find that the solution of the sstem is (s, t) (, ). Check this solution b substituting it into both equations. 78 Discovering Algebra Condensed Lessons Ke Curriculum Press
5 CONDENSED L E S S O N 6. Solving Sstems of Equations Using Elimination Previous Net In this lesson ou will represent situations with sstems of equations use the elimination method to solve sstems of linear equations Read the tet at the beginning of Lesson 6. in our book. It eplains that ou can add two equations to get another true equation. Then, read Eample A carefull and make sure ou understand it. In the eample, the variable s is eliminated just b adding the equations. As ou will see in the investigation, sometimes using the elimination method requires a bit more work. Investigation: Paper Clips and Pennies Place one paper clip along the long side of a piece of paper. Then, line up enough pennies to complete the 11-inch length. If ou use a jumbo paper clip, ou should find that ou need 1 pennies. Place two paper clips along the short side of the sheet of paper, and add pennies to complete the 8.5-inch length. With jumbo paper clips, ou ll need 6 pennies. If C is the length of a paper clip and P is the diameter of a penn, ou can write this sstem of equations to represent this situation. C 1P 11 C 6P 8.5 Notice that ou can t eliminate a variable b adding the two original equations. However, look what happens when ou multipl both sides of the first equation b. C 1P 11 C P C 6P 8.5 C 6P 8.5 Because ou multiplied both sides of the first equation b the same number, the new equation has the same solutions as the original. You can now eliminate the variable C b adding the two equations in the new sstem. C P C 6P 8.5 Long side Short side 18P 1.5 Add the equations. P.75 Divide b 18. To find the value of C, substitute.75 for P in either equation and solve for C. C 1(.75) 11 or C 6(.75) 8.5 (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 79
6 Previous Net Lesson 6. Solving Sstems of Equations Using Elimination (continued) You should find that C is. Be sure to check the solution b substituting.75 for P and for C in both equations. 1(.75) 11 and () 6(.75) 8.5 The solution (.75, ) means that the penn has a diameter of.75 inch and the paper clip has a length of inches. There are several was ou could have solved the original sstem of equations. For eample, instead of multipling the first equation b, ou could have multiplied the second equation b. Then, the coefficient of P would be 1 in both equations and ou could eliminate P b adding the equations. Read the rest of the lesson in our book. Here is an additional eample. EXAMPLE At Marli s Discount Music Mart, all CDs are the same price and all cassette tapes are the same price. Rashid bought si CDs and five cassette tapes for $ Quinc bought four CDs and nine cassette tapes for $1.7. Write and solve a sstem of equations to find the price of a CD and the price of a cassette tape. Solution If c is the price of a CD and t is the price of a tape, then the problem can be modeled with this sstem. 6c 5t c 9t 1.7 Rashid s purchase Quinc s purchase If ou multipl the first equation b and the second equation b, ou will be able to add the equations to eliminate c. 6c 5t c t 5.56 Multipl both sides b. c 9t 1.7 1c 7t t t 7.98 Multipl both sides b. Add the equations. Divide. To find the value of c, substitute 7.98 for t in either equation and solve for t. 6c 5t Original first equation. 6c 5(7.98) Substitute 7.98 for t. 6c c Multipl. Subtract 9.9 from both sides. c 1.98 Divide both sides b 6. Cassette tapes cost $7.98 and CDs cost $1.98. Be sure to check this solution b substituting it into both original equations. 8 Discovering Algebra Condensed Lessons Ke Curriculum Press
7 CONDENSED L E S S O N 6. Solving Sstems of Equations Using Matrices Previous Net In this lesson ou will represent situations with sstems of equations use matrices to solve sstems of linear equations You now know how to solve sstems of equations with tables and graphs and b using the substitution and elimination methods. You can also solve sstems of equations b using matrices. Pages 1 of our book eplain how to represent a sstem of equations with a matri and then use row operations to find the solution. Read this tet and Eample A carefull. Investigation: Diagonalization Consider this sstem of equations. Because the equations are in standard form, ou can represent the sstem with a matri. Write the numerals from the first equation in the first row, and write the numerals from the second equation in the second row. To solve the equation, perform row operations to get 1 s in the diagonal of the matri and s above and below the diagonal as shown here. To get a as the first entr in the second row, add times the first row to the second row. This step is similar to using the elimination method to eliminate from the second equation. times row 1 6 New matri row New row 8 To get 1 as the second entr in the second row, divide that row b a 1 b From the second row, ou can see that. Now, subtract the second row from the first to get a as the second entr in the first row. This is similar to substituting for in the first equation to get 8. Row New matri row 1 8 New row (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 81
8 Previous Net Lesson 6. Solving Sstems of Equations Using Matrices (continued) To get a 1 as the first entr in the first row, divide the row b. 1 1 You can now see that and. You can check this solution b substituting it into the original equation. Eample B in our book shows that matrices are useful for solving sstems of equations involving large numbers. Here is another eample. EXAMPLE At a college football game, students paid $1 per ticket and nonstudents paid $18 per ticket. The number of students who attended was 1, more than the number of nonstudents. The total of all ticket sales was $67,6. How man of the attendees were students, and how man were nonstudents? Solution If S is the number of students and N is the number of nonstudents, then ou can represent the situation with this sstem and matri. S N 1, 1 1 1, 1S 18N 67, ,6 Use row operations to find the solution. Add 1 times row 1 to row to get 1 1 1, new row. 5, 1 1 1, Divide row b. 1 1,67 1, Add row to row 1 to get new row ,67 The final matri shows that S, and N 1,67. So, students and 1,67 nonstudents attended the game. You can check this solution b substituting it into both original equations. 8 Discovering Algebra Condensed Lessons Ke Curriculum Press
9 CONDENSED L E S S O N 6.5 Inequalities in One Variable Previous Net In this lesson ou will write inequalities to represent situations learn how appling operations to both sides of an inequalit affects the direction of the inequalit smbol solve a problem b writing and solving an inequalit An inequalit is a statement that one quantit is less than or greater than another. Inequalities are written using the smbols,,, and. Read the tet on page 9 of our book, which gives several eamples from everda life and how to write them as inequalities. Just as with equations, ou can solve inequalities b appling the same operations to both sides. However, as ou will learn in the investigation, ou need to be careful about the direction of the inequalit smbol. Investigation: Toe the Line In this investigation, two walkers stand on a number line. Walker A starts on the number, and Walker B starts on the number. You can represent this situation with the inequalit. Steps 1 When an announcer calls out an operation, the walkers perform the operation on their numbers and move to new positions based on the result. The new positions are represented b an inequalit, with the position of Walker A on the left side and the position of Walker B on the right side. The drawings below show the walkers positions after the first two operations along with the corresponding inequalit. Operation: Add ; Inequalit: 6 Operation: Subtract ; Inequalit: 1 A B A B A B This table shows the results of the remaining operations. Walker A s Inequalit Walker B s Operation position smbol position Add 1 1 Subtract 5 Multipl b 6 Subtract 7 1 Multipl b 9 Add 5 8 Divide b 1 Subtract 1 Multipl b 1 1 (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 8
10 Previous Net Lesson 6.5 Inequalities in One Variable (continued) Steps 5 9 Notice that when a number is added to or subtracted from the walkers positions, the direction of the inequalit (that is, the relative positions of the walkers) remains the same. The direction of an inequalit also stas the same when the positions are multiplied or divided b a positive number. However, when the positions are multiplied or divided b a negative number, the direction of the inequalit (that is, the relative positions of the walkers) is reversed. Check these findings b starting with another inequalit and appling operations to both sides. You should find that the direction of the inequalit smbol is reversed onl when ou multipl or divide b a negative number. Read Eample A in our book, which shows how to graph solutions to inequalities on a number line. Then, read Eample B, which applies what ou learned in the investigation to solve an inequalit. Here is an additional eample. EXAMPLE A Jack takes the bus to the bowling alle. He has $15 when he arrives. It costs $.5 to bowl one game. If Jack needs $1.5 to take the bus home, how man games can he bowl? Solve this problem b writing and solving an inequalit. Solution Let g represent the number of games Jack can bowl. We know that the amount Jack starts with minus the amount he spends bowling must be at least (that is, greater than or equal to) $1.5. So we can write this inequalit. Amount Jack starts with Cost of bowling g games Bus fare 15.5g 1.5 Now, solve the inequalit. 15.5g 1.5 Original inequalit g Subtract 15 from both sides..5g g g 6 Subtract. Divide both sides b.5, and reverse the inequalit smbol. Divide. Jack can bowl 6 games or fewer. Here, g 6 is graphed on a number line Discovering Algebra Condensed Lessons Ke Curriculum Press
11 CONDENSED L E S S O N 6.6 Graphing Inequalities in Two Variables Previous Net In this lesson ou will graph linear inequalities in two variables You know how to graph linear equations in two variables, such as 6. In this lesson ou will learn to graph linear inequalities in two variables, such as 6 and 6. Investigation: Graphing Inequalities To complete this investigation, ou ll need a grid worksheet like the one on page 7 of our book. Choose one of the statements listed on page 7. For each point shown with a circle on the worksheet, substitute the coordinates of the point into the statement, and then fill in the circle with the relational smbol,,, or, that makes the statement true. For eample, if ou choose the statement 1, do the following for the point (, ): 1 Original statement. 1 () Substitute for and for. 7 Subtract. Because the smbol makes this statement true, write in the circle corresponding to the point (, ). Here are completed grids for the four statements (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 85
12 Previous Net Lesson 6.6 Graphing Inequalities in Two Variables (continued) Notice that for each statement, the circles containing equal signs form a straight line. All the circles above the line are filled in with smbols, and all the circles below the line are filled in with smbols. Choose one of the statements and test a point with fractional or decimal coordinates. For eample, in the grid for 1, (., 1.5) is below the line of equal signs. Substitute the coordinates into the statement. 1 Original statement (.) Substitute. for and 1.5 for Subtract Insert the appropriate smbol. The resulting statement gets a smbol, just like the other points below the line of equal signs. Shown here are graphs of 1, 1, 1, 1, and 1. In each graph the shaded regions include the points that make the statement true. A dashed line indicates that the line is not included in the graph. A solid line indicates that the line is included Make similar graphs for the other inequalities. You should notice the following: Graphs of inequalities in the form epression and epression are shaded above the line. Graphs of inequalities in the form epression and epression are shaded below the line. Graphs of inequalities in the form epression and epression require a solid line. Graphs of inequalities in the form epression and epression require a dashed line. Read the rest of the lesson and the eample in our book. When ou are finished, ou should be able to graph an linear inequalit. 86 Discovering Algebra Condensed Lessons Ke Curriculum Press
13 CONDENSED L E S S O N 6.7 Sstems of Inequalities Previous Net In this lesson ou will graph solutions of sstems of inequalities use sstems of inequalities to represent situations involving constraints You can find the solution of a sstem of equations b graphing the equations and locating the points of intersection. You can use a similar method to find the solution of a sstem of inequalities. Read Eample A in our book. Then, read the additional eample below. EXAMPLE Graph this sstem of inequalities and indicate the solution. Solution Graph with a solid line because its points satisf the inequalit. Shade above the line because its inequalit has the greater than or equal to smbol. Graph with a dashed line because its points do not satisf the inequalit. Shade below the line because in the inequalit is less than the epression in. The points in the overlapping region satisf both inequalities, so the overlapping region is the solution of the sstem. _ 6 Eample B in our book shows how sstems of inequalities are useful for modeling situations involving constraints. Read through the eample. Investigation: A Tpical Envelope Here are two constraints the U.S. Postal Service imposes on envelope sizes. The ratio of length to width must be less than or equal to.5. The ratio of length to width must be greater than or equal to 1.. If l and w represent the length and width of an envelope, then the first constraint l can be represented b the equation w.5 and the second can be represented l b w 1.. You can solve each inequalit for l b multipling both sides b w. This gives the sstem l.5w l 1.w (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 87
14 Previous Net Lesson 6.7 Sstems of Inequalities (continued) Note that ou do not need to reverse the direction of the inequalit smbol when ou multipl both sides b because the width of an envelope must be a positive number. Here, both inequalities are graphed on the same aes. The overlap of the shaded regions is the solution of the sstem. You can check this b choosing a point from the overlapping region and making sure its coordinates satisf both inequalities. Length (in.) 5 l Step 5 in our book gives the dimensions of four envelopes. Points corresponding to these envelopes are plotted on the graph here. Point a, which corresponds to a 5 in.-b-8 in. envelope, and Point d, which corresponds to a 5.5 in.-b-7.5 in. envelope, fall within the overlapping regions, indicating that these envelopes satisf both constraints. Notice that (, ) satisfies the sstem. This point corresponds to an envelope with no length or width, which does not make sense. Adding constraints specifing minimum and maimum lengths and widths would make the sstem a more realistic model. For eample, for an envelope to require a stamp, the length must be between 5 in. and 11.5 in. and the width must be between.5 in. and 6.15 in. The sstem includes these constraints and has this graph. l.5w l 1.w l 5 w.5 l 11.5 w 6.15 Length (in.) l w.5 Length (in.) 5 l c. b. w Width (in.) a. d. 5 Width (in.) w w l 11.5 l Width (in.) w 88 Discovering Algebra Condensed Lessons Ke Curriculum Press
5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationSolution of the System of Linear Equations: any ordered pair in a system that makes all equations true.
Definitions: Sstem of Linear Equations: or more linear equations Sstem of Linear Inequalities: or more linear inequalities Solution of the Sstem of Linear Equations: an ordered pair in a sstem that makes
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationWhy should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationSolving Absolute Value Equations and Inequalities Graphically
4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More informationLinear Inequality in Two Variables
90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More informationSystems of Linear Equations: Solving by Substitution
8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing
More information{ } Sec 3.1 Systems of Linear Equations in Two Variables
Sec.1 Sstems of Linear Equations in Two Variables Learning Objectives: 1. Deciding whether an ordered pair is a solution.. Solve a sstem of linear equations using the graphing, substitution, and elimination
More informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -
More informationSolving Special Systems of Linear Equations
5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest
More informationSkills Practice Skills Practice for Lesson 1.1
Skills Practice Skills Practice for Lesson. Name Date Tanks a Lot Introduction to Linear Functions Vocabular Define each term in our own words.. function A function is a relation that maps each value of
More informationSYSTEMS OF LINEAR EQUATIONS
SYSTEMS OF LINEAR EQUATIONS Sstems of linear equations refer to a set of two or more linear equations used to find the value of the unknown variables. If the set of linear equations consist of two equations
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
1. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationFunctions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationEQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM
. Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
More informationZero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m
0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationLINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,
LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationCHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS
CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRE-CALCULUS: A TEACHING TEXTBOOK Lesson 64 Solving Sstems In this chapter, we re going to focus on sstems of equations. As ou ma remember from algebra, sstems
More information1. 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
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 informationSolving Systems of Equations
Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationFor 14 15, use the coordinate plane shown. represents 1 kilometer. 10. Write the ordered pairs that represent the location of Sam and the theater.
Name Class Date 12.1 Independent Practice CMMN CRE 6.NS.6, 6.NS.6b, 6.NS.6c, 6.NS.8 m.hrw.com Personal Math Trainer nline Assessment and Intervention For 10 13, use the coordinate plane shown. Each unit
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationLinear Equations in Two Variables
Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations
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 information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph
More informationName Class Date. Additional Vocabulary Support
- Additional Vocabular Support Rate of Change and Slope Concept List negative slope positive slope rate of change rise run slope slope formula slope of horizontal line slope of vertical line Choose the
More informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationSummer Math Exercises. For students who are entering. Pre-Calculus
Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationMath 152, Intermediate Algebra Practice Problems #1
Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations
More informationExample 1: Model A Model B Total Available. Gizmos. Dodads. System:
Lesson : Sstems of Equations and Matrices Outline Objectives: I can solve sstems of three linear equations in three variables. I can solve sstems of linear inequalities I can model and solve real-world
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationNAME DATE PERIOD. 11. Is the relation (year, percent of women) a function? Explain. Yes; each year is
- NAME DATE PERID Functions Determine whether each relation is a function. Eplain.. {(, ), (0, 9), (, 0), (7, 0)} Yes; each value is paired with onl one value.. {(, ), (, ), (, ), (, ), (, )}. No; in the
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationSome Tools for Teaching Mathematical Literacy
Some Tools for Teaching Mathematical Literac Julie Learned, Universit of Michigan Januar 200. Reading Mathematical Word Problems 2. Fraer Model of Concept Development 3. Building Mathematical Vocabular
More information1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) =
Extra Practice for Lesson Add or subtract. ) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = Multiply. 7) (5)(-4) = 8) (-3)(-6) = 9) (-)(2) = Division is
More informationWe start with the basic operations on polynomials, that is adding, subtracting, and multiplying.
R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract
More informationRotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012
Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to
More informationChapter 3 & 8.1-8.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x - 4y = 10 16x + 8y = 10
Chapter 3 & 8.1-8.3 These are meant for practice. The actual test is different. Determine whether the pair of equations represents parallel lines. 1) 9 + 3 = 12 27 + 9 = 39 1) Determine whether the pair
More informationRELEASED. North Carolina READY End-of-Grade Assessment Mathematics. Grade 8. Student Booklet
REVISED 7/4/205 Released Form North Carolina READY End-of-Grade Assessment Mathematics Grade 8 Student Booklet Academic Services and Instructional Support Division of Accountabilit Services Copright 203
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationALGEBRA 1 SKILL BUILDERS
ALGEBRA 1 SKILL BUILDERS (Etra Practice) Introduction to Students and Their Teachers Learning is an individual endeavor. Some ideas come easil; others take time--sometimes lots of time- -to grasp. In addition,
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationClassifying Solutions to Systems of Equations
CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Classifing Solutions to Sstems of Equations Mathematics Assessment Resource Service Universit of Nottingham
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationAlgebra I. Practice Test 2 Booklet. Student Name:
Student Name: lgebra I Practice Test ooklet This publication/document has been produced under a contract with the Mississippi Department of Education. Neither the Department nor an other entities, public
More information135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the
More information1 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
More informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationMath Questions & Answers
What five coins add up to a nickel? five pennies (1 + 1 + 1 + 1 + 1 = 5) Which is longest: a foot, a yard or an inch? a yard (3 feet = 1 yard; 12 inches = 1 foot) What do you call the answer to a multiplication
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 informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationUnit 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
More informationAmerican Diploma Project
Student Name: American Diploma Project ALGEBRA l End-of-Course Eam PRACTICE TEST General Directions Today you will be taking an ADP Algebra I End-of-Course Practice Test. To complete this test, you will
More informationPart 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
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 24, 2012 9:15 a.m. to 12:15 p.m.
INTEGRATED ALGEBRA The Universit of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesda, Januar 4, 01 9:15 a.m. to 1:15 p.m., onl Student Name: School Name: Print our name and
More informationAssessment Anchors and Eligible Content
M07.A-N The Number System M07.A-N.1 M07.A-N.1.1 DESCRIPTOR Assessment Anchors and Eligible Content Aligned to the Grade 7 Pennsylvania Core Standards Reporting Category Apply and extend previous understandings
More informationHow many of these intersection points lie in the interior of the shaded region? If 1. then what is the value of
NOVEMBER A stack of 00 nickels has a height of 6 inches What is the value, in dollars, of an 8-foot-high stack of nickels? Epress our answer to the nearest hundredth A cube is sliced b a plane that goes
More information2 Solving Systems of. Equations and Inequalities
Solving Sstems of Equations and Inequalities. Solving Linear Sstems Using Substitution. Solving Linear Sstems Using Elimination.3 Solving Linear Sstems Using Technolog.4 Solving Sstems of Linear Inequalities
More informationMathematical goals. Starting points. Materials required. Time needed
Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between
More informationSTUDENT TEXT AND HOMEWORK HELPER
UNIT 4 EXPONENTIAL FUNCTIONS AND EQUATIONS STUDENT TEXT AND HOMEWORK HELPER Randall I. Charles Allan E. Bellman Basia Hall William G. Handlin, Sr. Dan Kenned Stuart J. Murph Grant Wiggins Boston, Massachusetts
More information5 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
More information