Lesson Menu Five-Minute Check (over Chapter 2) CCSS Then/Now New Vocabulary Example 1:Solve by Using a Table Example 2:Solve by Graphing Example 3:Classify Systems Concept Summary: Characteristics of Linear Systems Key Concept: Substitution Method Example 4:Real-World Example: Use the Substitution Method Key Concept: Elimination Method Example 5: Solve by Using Elimination Example 6: Standardized Test Example: No Solution and Infinite Solutions Concept Summary: Solving Systems of Equations

Over Chapter 2 5-Minute Check 1 A.D = {–4, –2, 0, 1, 2}, R = {–4, 0,1}; yes B.D = {0, 1, 2}, R = {0, 1}; yes C.D = {–4, 0, 1}, R = {–4, –2, 0, 1, 2}; no D.D = {–2, –4}; R = {–4, 0, 1}; yes Find the domain and range of the relation {(–4, 1), (0, 0), (1, –4), (2, 0), (–2, 0)}. Determine whether the relation is a function.

Over Chapter 2 5-Minute Check 1 A.D = {–4, –2, 0, 1, 2}, R = {–4, 0,1}; yes B.D = {0, 1, 2}, R = {0, 1}; yes C.D = {–4, 0, 1}, R = {–4, –2, 0, 1, 2}; no D.D = {–2, –4}; R = {–4, 0, 1}; yes Find the domain and range of the relation {(–4, 1), (0, 0), (1, –4), (2, 0), (–2, 0)}. Determine whether the relation is a function.

Over Chapter 2 5-Minute Check 2 A.28 B.12 C.–12 D.–16 Find the value of f(4) for f(x) = 8 – x – x 2.

Over Chapter 2 5-Minute Check 2 A.28 B.12 C.–12 D.–16 Find the value of f(4) for f(x) = 8 – x – x 2.

Over Chapter 2 5-Minute Check 3 Find the slope of the line that passes through (5, 7) and (–1, 0). A. B. C.2 D.7

Over Chapter 2 5-Minute Check 3 Find the slope of the line that passes through (5, 7) and (–1, 0). A. B. C.2 D.7

Over Chapter 2 5-Minute Check 4 A.y = –3x + 6 B.y = –3x – 6 C.y = 3x + 6 D.y = 2x + 6 Write an equation in slope-intercept form for the line that has x-intercept –3 and y-intercept 6.

Over Chapter 2 5-Minute Check 4 A.y = –3x + 6 B.y = –3x – 6 C.y = 3x + 6 D.y = 2x + 6 Write an equation in slope-intercept form for the line that has x-intercept –3 and y-intercept 6.

Over Chapter 2 5-Minute Check 5 A.15 B.16 C.17 D.18 The Math Club is using the prediction equation y = 1.25x + 10 to estimate the number of members it will have, where x represents the number of years the club has been in existence. About how many members does the club expect to have in its fifth year?

Over Chapter 2 5-Minute Check 5 A.15 B.16 C.17 D.18 The Math Club is using the prediction equation y = 1.25x + 10 to estimate the number of members it will have, where x represents the number of years the club has been in existence. About how many members does the club expect to have in its fifth year?

Over Chapter 2 5-Minute Check 6 A.absolute value B.linear C.piecewise-defined D.quadratic Identify the type of function represented by the equation y = 4x

Over Chapter 2 5-Minute Check 6 A.absolute value B.linear C.piecewise-defined D.quadratic Identify the type of function represented by the equation y = 4x

Then/Now You graphed and solved linear equations. Solve systems of linear equations graphically. Solve systems of linear equations algebraically.

Vocabulary break-even point system of equations consistent inconsistent independent dependent substitution method elimination method

Example 1 What is the solution of the system of equations? x + y = 2 x – 3y = –6 A.(1, 1) B.(0, 2) C.(2, 0) D.(–4, 6)

Example 1 What is the solution of the system of equations? x + y = 2 x – 3y = –6 A.(1, 1) B.(0, 2) C.(2, 0) D.(–4, 6)

Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation in slope-intercept form.

Example 2 Solve by Graphing Check Substitute the coordinates into each equation. x – 2y= 0x + y=6Original equations 4 – 2(2)= =6Replace x with 4 and y with 2. ?? 0=06=6Simplify. Answer:

Example 2 Solve by Graphing Check Substitute the coordinates into each equation. x – 2y= 0x + y=6Original equations 4 – 2(2)= =6Replace x with 4 and y with 2. ?? 0=06=6Simplify. Answer: The solution of the system is (4, 2).

Example 2 Which graph shows the solution to the system of equations below? x + 3y=7 x – y=3 A.C. B.D.

Example 2 Which graph shows the solution to the system of equations below? x + 3y=7 x – y=3 A.C. B.D.

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Example 4 Use the Substitution Method FURNITURE Lancaster Woodworkers Furniture Store builds two types of wooden outdoor chairs. A rocking chair sells for $265 and an Adirondack chair with footstool sells for $320. The books show that last month, the business earned $13,930 for the 48 outdoor chairs sold. How many of each chair were sold? Understand You are asked to find the number of each type of chair sold.

Example 4 Use the Substitution Method Define variables and write the system of equations. Let x represent the number of rocking chairs sold and y represent the number of Adirondack chairs sold. x + y =48The total number of chairs sold was x + 320y =13,930The total amount earned was $13,930. Plan

Example 4 Use the Substitution Method Solve one of the equations for one of the variables in terms of the other. Since the coefficient of x is 1, solve the first equation for x in terms of y. x + y =48First equation x=48 – ySubtract y from each side.

Example 4 Use the Substitution Method Solve Substitute 48 – y for x in the second equation. 265x + 320y =13,930Second equation 265(48 – y) + 320y =13,930Substitute 48 – y for x. 12,720 – 265y + 320y=13,930Distributive Property 55y=1210Simplify. y=22Divide each side by 55.

Example 4 Use the Substitution Method Now find the value of x. Substitute the value for y into either equation. x + y =48First equation x + 22 =48Replace y with 22. x=26Subtract 22 from each side. Answer:

Example 4 Use the Substitution Method Now find the value of x. Substitute the value for y into either equation. x + y =48First equation x + 22 =48Replace y with 22. x=26Subtract 22 from each side. Answer:They sold 26 rocking chairs and 22 Adirondack chairs.

Example 4 A.210 adult; 120 children B.120 adult; 210 children C.300 children; 30 adult D.300 children; 30 adult AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24.50 for adults and $16.50 for children. On Sunday, the amusement park made $6405 from selling 330 tickets. How many of each kind of ticket was sold?

Example 4 A.210 adult; 120 children B.120 adult; 210 children C.300 children; 30 adult D.300 children; 30 adult AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24.50 for adults and $16.50 for children. On Sunday, the amusement park made $6405 from selling 330 tickets. How many of each kind of ticket was sold?

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Example 5 Solve by Using Elimination Use the elimination method to solve the system of equations. x + 2y = 10 x + y = 6 In each equation, the coefficient of x is 1. If one equation is subtracted from the other, the variable x will be eliminated. x + 2y=10 (–)x + y= 6 y= 4Subtract the equations.

Example 5 Solve by Using Elimination Now find x by substituting 4 for y in either original equation. x + y=6Second equation x + 4=6Replace y with 4. x= 2Subtract 4 from each side. Answer:

Example 5 Solve by Using Elimination Now find x by substituting 4 for y in either original equation. x + y=6Second equation x + 4=6Replace y with 4. x= 2Subtract 4 from each side. Answer:The solution is (2, 4).

Example 5 A.(2, –1) B.(17, –4) C.(2, 1) D.no solution Use the elimination method to solve the system of equations. What is the solution to the system? x + 3y = 5 x + 5y = –3

Example 5 A.(2, –1) B.(17, –4) C.(2, 1) D.no solution Use the elimination method to solve the system of equations. What is the solution to the system? x + 3y = 5 x + 5y = –3

Example 6 No Solution and Infinite Solutions Read the Test Item You are given a system of two linear equations and are asked to find the solution. Solve the system of equations. 2x + 3y = 12 5x – 2y = 11 A. (2, 3) B. (6, 0) C. (0, 5.5) D. (3, 2)

Example 6 No Solution and Infinite Solutions x =3x =3 Solve the Test Item Multiply the first equation by 2 and the second equation by 3. Then add the equations to eliminate the y variable. 2x + 3y=124x + 6y=24 Multiply by 2. Multiply by 3. 5x – 2y=11(+)15x – 6y=33 19x =57

Example 6 No Solution and Infinite Solutions Replace x with 3 and solve for y. 2x + 3y=12First equation 2(3) + 3y=12Replace x with y=12Multiply. 3y=6Subtract 6 from each side. y=2Divide each side by 3. Answer:

Example 6 No Solution and Infinite Solutions Replace x with 3 and solve for y. 2x + 3y=12First equation 2(3) + 3y=12Replace x with y=12Multiply. 3y=6Subtract 6 from each side. y=2Divide each side by 3. Answer:The solution is (3, 2). The correct answer is D.

Example 6 Solve the system of equations. x + 3y = 7 2x + 5y = 10 A. B.(1, 2) C.(–5, 4) D.no solution

Example 6 Solve the system of equations. x + 3y = 7 2x + 5y = 10 A. B.(1, 2) C.(–5, 4) D.no solution

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