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 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. 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 Over Chapter 2 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. 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 Over Chapter 2 Find the value of f(4) for f(x) = 8 – x – x2. A. 28 B. 12 C. –12 D. –16 Over Chapter 2 Find the value of f(4) for f(x) = 8 – x – x2. A. 28 B. 12 C. –12 D. –16 Over Chapter 2 Find the slope of the line that passes through (5, 7) and (–1, 0). A. B. C. 2 D. 7 Over Chapter 2 Find the slope of the line that passes through (5, 7) and (–1, 0). A. B. C. 2 D. 7 Over Chapter 2 Write an equation in slope-intercept form for the line that has x-intercept –3 and y-intercept 6. A. y = –3x + 6 B. y = –3x – 6 C. y = 3x + 6 D. y = 2x + 6 Over Chapter 2 Write an equation in slope-intercept form for the line that has x-intercept –3 and y-intercept 6. A. y = –3x + 6 B. y = –3x – 6 C. y = 3x + 6 D. y = 2x + 6 Over Chapter 2 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? A. 15 B. 16 C. 17 D. 18 Over Chapter 2 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? A. 15 B. 16 C. 17 D. 18 Over Chapter 2 Identify the type of function represented by the equation y = 4x2 + 6. A. absolute value B. linear C. piecewise-defined D. quadratic Over Chapter 2 Identify the type of function represented by the equation y = 4x2 + 6. A. absolute value B. linear C. piecewise-defined D. quadratic You graphed and solved linear equations. • Solve systems of linear equations graphically. • Solve systems of linear equations algebraically. • break-even point • independent • system of equations • dependent • consistent • substitution method • inconsistent • elimination method 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) 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) Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x+y=6 Write each equation in slope-intercept form. The graphs appear to intersect at (4, 2). Solve by Graphing Check Substitute the coordinates into each equation. x – 2y = 0 ? x+y =6 ? 4 – 2(2) = 0 4+2 =6 0=0 6=6 Answer: Original equations Replace x with 4 and y with 2. Simplify. Solve by Graphing Check Substitute the coordinates into each equation. x – 2y = 0 ? x+y =6 ? 4 – 2(2) = 0 4+2 =6 0=0 6=6 Original equations Replace x with 4 and y with 2. Simplify. Answer: The solution of the system is (4, 2). Which graph shows the solution to the system of equations below? x + 3y = 7 x–y = 3 A. C. B. D. Which graph shows the solution to the system of equations below? x + 3y = 7 x–y = 3 A. C. B. D. 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. Use the Substitution Method Plan 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 = 48 265x + 320y = 13,930 The total number of chairs sold was 48. The total amount earned was $13,930. 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 = 48 x = 48 – y First equation Subtract y from each side. Use the Substitution Method Solve Substitute 48 – y for x in the second equation. 265x + 320y = 13,930 265(48 – y) + 320y = 13,930 12,720 – 265y + 320y = 13,930 55y = 1210 y = 22 Second equation Substitute 48 – y for x. Distributive Property Simplify. Divide each side by 55. Use the Substitution Method Now find the value of x. Substitute the value for y into either equation. x + y = 48 x + 22 = 48 x = 26 Answer: First equation Replace y with 22. Subtract 22 from each side. Use the Substitution Method Now find the value of x. Substitute the value for y into either equation. x + y = 48 x + 22 = 48 x = 26 First equation Replace y with 22. Subtract 22 from each side. Answer: They sold 26 rocking chairs and 22 Adirondack chairs. 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? 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? A. 210 adult; 120 children B. 120 adult; 210 children C. 300 children; 30 adult D. 300 children; 30 adult 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= 4 Subtract the equations. Solve by Using Elimination Now find x by substituting 4 for y in either original equation. x+y =6 Second equation x+4 =6 Replace y with 4. x =2 Answer: Subtract 4 from each side. Solve by Using Elimination Now find x by substituting 4 for y in either original equation. x+y =6 Second equation x+4 =6 Replace y with 4. x =2 Subtract 4 from each side. Answer: The solution is (2, 4). Use the elimination method to solve the system of equations. What is the solution to the system? x + 3y = 5 x + 5y = –3 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 A. (2, –1) B. (17, –4) C. (2, 1) D. no solution No Solution and Infinite Solutions Solve the system of equations. 2x + 3y = 12 5x – 2y = 11 A. (2, 3) C. (0, 5.5) B. (6, 0) D. (3, 2) Read the Test Item You are given a system of two linear equations and are asked to find the solution. No Solution and Infinite Solutions 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 = 12 Multiply by 2. 5x – 2y = 11 4x + 6y = 24 (+)15x – 6y = 33 19x = 57 Multiply by 3. x=3 No Solution and Infinite Solutions Replace x with 3 and solve for y. 2x + 3y = 12 2(3) + 3y = 12 6 + 3y = 12 3y = 6 y=2 Answer: First equation Replace x with 3. Multiply. Subtract 6 from each side. Divide each side by 3. No Solution and Infinite Solutions Replace x with 3 and solve for y. 2x + 3y = 12 2(3) + 3y = 12 6 + 3y = 12 3y = 6 y=2 First equation Replace x with 3. Multiply. Subtract 6 from each side. Divide each side by 3. Answer: The solution is (3, 2). The correct answer is D. Solve the system of equations. x + 3y = 7 2x + 5y = 10 A. B. (1, 2) C. (–5, 4) D. no solution Solve the system of equations. x + 3y = 7 2x + 5y = 10 A. B. (1, 2) C. (–5, 4) D. no solution