Splash Screen

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

Over Lesson 3–1 5-Minute Check 1 Solve the system of equations y = 3x – 2 and y = –3x + 2 by graphing. A. B.(1, –1) C. D.(–1, 1)

Over Lesson 3–1 5-Minute Check 2 A.consistent and independent B.consistent and dependent C.inconsistent Graph the system of equations 2x + y = 6 and 3y = –6x + 6. Describe it as consistent and independent, consistent and dependent, or inconsistent.

Over Lesson 3–1 5-Minute Check 3 A.5 multiple choice, 25 true/false B.10 multiple choice, 20 true/false C.15 multiple choice, 15 true/false D.20 multiple choice, 10 true/false A test has 30 questions worth a total of 100 points. Each multiple choice question is worth 4 points and each true/false question is worth 3 points. How many of each type of question are on the test?

Then/Now You solved systems of linear equations by using tables and graphs. Solve systems of linear equations by using substitution. Solve systems of linear equations by using elimination.

Vocabulary substitution method elimination method

Concept

Example 1 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 1 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 1 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 1 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 1 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 1 Use the Substitution Method Check You can use a graphing calculator to check this solution.

Example 1 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?

Concept

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

Example 4 No Solution and Infinite Solutions A. Use the elimination method to solve the system of equations. –3x + 5y = 12 6x – 10y = –21 Use multiplication to eliminate x. –3x + 5y=12–6x + 10y= 24 Multiply by 2. 0 = 3 6x – 10y=–21(+)6x – 10y=–21 Answer: Since there are no values of x and y that will make the equation 0 = 3 true, there are no solutions for the system of equations.

Example 4 No Solution and Infinite Solutions B. Use the elimination method to solve the system of equations. –3x + 4y = 7 9x – 12y = –21 Use multiplication to eliminate x. –3x + 4y=7–9x + 12y= 21 Multiply by 3. 0 = 0 9x – 12y=–21(+)9x – 12y=–21 Answer: Because the equation 0 = 0 is always true, there are an infinite number of solutions.

Example 4 A.(1, 3) B.(–5, 0) C.(2, –2) D.no solution Use the elimination method to solve the system of equations. What is the solution to the system of equations? 2x + 3y = 11 –4x – 6y = 20

Concept

End of the Lesson