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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
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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)
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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.
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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?
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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.
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Vocabulary substitution method elimination method
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Concept
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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.
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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 48. 265x + 320y =13,930The total amount earned was $13,930. Plan
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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.
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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.
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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.
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Example 1 Use the Substitution Method Check You can use a graphing calculator to check this solution.
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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?
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Concept
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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.
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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).
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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
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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)
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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
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Example 3 Replace x with 3 and solve for y. 2x + 3y=12First equation 2(3) + 3y=12Replace x with 3. 6 + 3y=12Multiply. 3y=6Subtract 6 from each side. y=2Divide each side by 3. Answer:The solution is (3, 2). The correct answer is D.
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Example 3 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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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.
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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.
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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
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Concept
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End of the Lesson
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