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Warm Up

Warm Up. Problem of the Day. Lesson Presentation. Lesson Quizzes. 3 V. = A. 1. C – S. h. 3. t. Warm Up Solve for the indicated variable. 1. P = R – C for R 2. V = Ah for A 3. R = for C. R = P + C. Rt + S = C. Problem of the Day

felicity-hopkins
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Warm Up

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  1. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

  2. 3V = A 1 C – S h 3 t Warm Up Solve for the indicated variable. 1.P = R – C for R 2.V = Ah for A 3.R = for C R = P + C Rt + S = C

  3. Problem of the Day At an audio store, stereos have 2 speakers and home-theater systems have 5 speakers. There are 30 sound systems with a total of 99 speakers. How many systems are stereo systems and how many are home-theater systems? 17 stereo systems, 13 home-theater systems

  4. Sunshine State Standards MA.8.A.1.3 Use tables, graphs, and models to represent, analyze, and solve real-world problems related to systems of linear equations.

  5. Caution! When solving systems of equations, remember to find values for all of the variables.

  6. Additional Example 1A: Solving Systems of Equations Solve the system of equations. y = 4x – 6 y = x + 3 The expressions x + 3 and 4x – 6 both equal y. So by the Transitive Property they are equal to each other. y= 4x – 6 y =x + 3 4x – 6 = x + 3

  7. Additional Example 1A Continued Solve the equation to find x. 4x – 6 = x + 3 – x– x Subtract x from both sides. 3x – 6 = 3 + 6+ 6 Add 6 to both sides. 3x9 Divide both sides by 3. 3 = 3 x = 3 To find y, substitute 3 for x in one of the original equations. y = x + 3 = 3 + 3 = 6 The solution is (3, 6).

  8. Additional Example 1B: Solving Systems of Equations y = 2x + 9 y = –8 + 2x 2x + 9 = –8 + 2x Transitive Property Subtract 2x from both sides. – 2x– 2x 9 ≠ –8 The system of equations has no solution.

  9. Check It Out: Example 1A Solve each system of equations. y = x – 5 y = 2x – 8 x – 5 = 2x – 8 y = x – 5 3 = x y = (3) – 5 = –2 The solution is (3, –2).

  10. Check It Out: Example 1B Solve each system of equations. y = 2x y = x + 6 2x = x + 6 y = 2x x = 6 y = 2(6) = 12 The solution is (6, 12).

  11. When equations in a system are not already solve for one variable, you can solve both equations for x or both for y.

  12. Additional Example 2A: Solving Systems of Equations by Solving for a Variable Solve the system of equations. x + 4y = –10 x – 3y = 11 Solve both equations for x. x + 4y = –10 x – 3y = 11 –4y–4y+ 3y+ 3y x = –10 – 4y x = 11 + 3y –10 – 4y = 11 + 3y –10 – 4y = 11 + 3y Subtract 3y from both sides. – 3y– 3y –10 – 7y = 11

  13. Additional Example 2A Continued –10 – 7y = 11 Add 10 to both sides. +10+10 – 7y 21 Divide both sides by –7. –7 = –7 y = –3 x = 11 + 3y = 11 + 3(–3)Substitute –3 for y. = 11 + –9 = 2 The solution is (2, –3).

  14. Helpful Hint You can solve for either variable. It is usually easiest to solve for a variable that has a coefficient of 1.

  15. = – –8 –2x –2 10y –2 –2 Additional Example 2B: Solving Systems of Equations by Solving for a Variable Solve the system of equations. –2x + 10y = –8 x – 5y = 4 Solve both equations for x. –2x + 10y = –8 x – 5y = 4 –10y–10y+5y+5y –2x = –8 – 10yx = 4 + 5y x = 4 + 5y Subtract 5y from both sides. 4 + 5y = 4 + 5y – 5y– 5y 4 = 4 Since 4 = 4 is always true, the system of equations has an infinite number of solutions.

  16. Check It Out: Example 2A Solve each system of equations. 2x + y = 0 2x + 3y = 8 −2x + 83 y = −2x and y = −2x + 83 –2x = ; x –2 2x + y = 0 2(−2) + y = 0 y = 4 The solution is (–2, 4).

  17. Check It Out: Example 2B Solve the system of equations. y = x –1 –3x + 3y = 4 3x + 43 y = x − 1 and y = 3x + 43 x – 1 = −3 ≠ 4 There are no solutions.

  18. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

  19. 1 2 ( , 2) Lesson Quiz Solve each system of equations. 1. y = 5x + 10 y = –7 + 5x 2.y = 2x + 1 y = 4x 3. 6x – y = –15 2x + 3y = 5 4. Two numbers have a sum of 23 and a difference of 7. Find the two numbers. no solution (–2, 3) 15 and 8

  20. Lesson Quiz for Student Response Systems 1. Solve the given system of equations. y = 11x + 20 y = –2 + 11x A. (2, 2) B. (1, 1) C. (1, –1) D. no solution

  21. Lesson Quiz for Student Response Systems 2. Solve the given system of equations. 4x + y = 11 2x + 3y = –7 A. (4, –5) B. (4, 5) C. (2, –5) D. (2, 5)

  22. Lesson Quiz for Student Response Systems 3. Two numbers have a sum of 37 and a difference of 17. Identify the two numbers. A. –27and–10 B. –27and10 C. 27 and 10 D. 27 and –10

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