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Equations in Two Variables: Solving and Graphing

Learn to solve and write solutions of equations in two variables as ordered pairs. Practice graphing and creating tables of solutions. Apply equations to real-life consumer math problems.

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Equations in Two Variables: Solving and Graphing

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

  2. Warm Up Solve. 1.x – 8 = 19 x = 27 a = 7 2. 5 = a – 2 n = 17 3. 7 + n = 24 4. 3c – 7 = 32 c = 13 y = 3 5. 17y + 7 = 58

  3. Problem of the Day A moving van travels 50 miles per hour. Use the equation y = 50x, where x represents the number of hours. How far will the van travel in 4.5 hours? 225 miles

  4. Learn to write solutions of equations in two variables as ordered pairs.

  5. Vocabulary ordered pair

  6. The company that makes team uniforms for a soccer league charges a $20 fee for team artwork and $10 for each jersey. Dominic’s team has 14 players, and Alyssa’s team has 12 players. Find the cost for a set of jerseys for each team. Let y be the total cost of a set of jerseys and x be the number of jerseys needed.

  7. $20 $10 # of jerseys total cost of jerseys • + = y = $20 + $10•x Dominic’s team: y = $20 + ($10 • 14)‏ y = $160 Alyssa’s team: y = $20 + ($10 • 12)‏ y = $140

  8. An ordered pair (x, y) is a pair of numbers that can be used to locate a point on a coordinate plane. A solution of a two-variable equation can be written as an ordered pair. The ordered pair (14, 160) is a solution because 160 = $20 + ($10 • 14). The ordered pair (12, 140) is a solution because 140 = $20 + ($10 • 12).

  9. ? 11 = 4(3) – 1 11 = 11 Helpful Hint The order in which a solution is written is important. Always write x first, then y. Additional Example 1A: Deciding Whether an Ordered Pair Is a Solution of an Equation Determine whether each ordered pair is a solution of y = 4x – 1. (3, 11)‏ y = 4x – 1 Substitute 3 for x and 11 for y. Simplify.  (3, 11) is a solution.

  10. ? 3 = 4(10) – 1 3 ≠ 39 Additional Example 1B: Deciding Whether an Ordered Pair Is a Solution of an Equation Determine whether each ordered pair is a solution of y = 4x – 1. (10, 3)‏ y = 4x – 1 Substitute 10 for x and 3 for y.  Simplify. (10, 3) is not a solution.

  11. ? 38 = 5(7) + 3 ? 38 = 38 Check It Out: Example 1A Determine whether each ordered pair is a solution of y = 5x + 3. (7, 38)‏ y = 5x + 3 Substitute 7 for x and 38 for y.  (7, 38) is a solution.

  12. ? 17 = 5(9) + 3 ? 17 = 48 Check It Out: Example 1B Determine whether each ordered pair is a solution of y = 5x + 3. (9, 17)‏ y = 5x + 3 Substitute 9 for x and 17 for y.  (9, 17) is not a solution.

  13. Helpful Hint 1 A table of solutions can be set up vertically or horizontally. 2 3 4 Additional Example 2A: Creating a Table of Ordered Pair Solutions Use the given values to make a table of solutions. y = x + 3for x = 1, 2, 3, 4 x x + 3 y (x, y)‏ 1 + 3 4 (1, 4)‏ 2 + 3 5 (2, 5)‏ 3 + 3 6 (3, 6)‏ 4 + 3 7 (4, 7)‏

  14. m 1 2 3 6m – 5 n (m, n)‏ Additional Example 2B: Creating a Table of Ordered Pair Solutions Use the given values to make a table of solutions. n = 6m –5for m = 1, 2, 3 6(1) – 5 6(2) – 5 6(3) – 5 1 7 13 (2, 7)‏ (1, 1)‏ (3, 13)‏

  15. 1 2 3 4 Check It Out: Example 2A Use the given values to make a table of solutions. y = x + 6for x = 1, 2, 3, 4 x x + 6 y (x, y)‏ 1 + 6 7 (1, 7)‏ 2 + 6 8 (2, 8)‏ 3 + 6 9 (3, 9)‏ 4 + 6 10 (4, 10)‏

  16. m 1 2 3 4 8m – 2 n (m, n)‏ Check It Out: Example 2B Use the given values to make a table of solutions. n = 8m –2for m = 1, 2, 3, 4 8(4) – 2 8(1) – 2 8(2) – 2 8(3) – 2 6 14 22 30 (2, 14)‏ (4, 30)‏ (1, 6)‏ (3, 22)‏

  17. Additional Example 3A: ConsumerMath Application A salesman marks up the price of everything he sells by 20%. The equation for the sales price p is p = 1.2w, where wis wholesale cost. What will be the sales price of a sweater with a wholesale cost of $48? p = 1.2(48) The wholesale cost of the sweater before tax is $48. Multiply. p = 57.6 The $48 wholesale sweater will cost the customer $57.60, so (48, 57.60) is a solution of the equation.

  18. Additional Example 3B: Consumer Math Application A salesman marks up the price of everything he sells by 20%. The equation for the sales price p is p = 1.2w, where wis wholesale cost. What will be the sales price of a jacket with a wholesale cost of $85? p = 1.2(85) The wholesale cost of the jacket before tax is $85. Multiply. p = 102 The $85.00 wholesale jacket will cost the customer $102, so (85, 102) is a solution of the equation.

  19. Check It Out: Example 3A In most states, the price of each item is not the total cost. Sales tax must be added. If sales tax is 7.5%, the equation for total cost is c = 1.075p, where p is the price before tax. How much will a $22 item cost after sales tax? c = 1.075(22) The price of the item before tax is $22. Multiply. c = 23.65 After sales tax, the $22 item will cost $23.65, so (22, 23.65) is a solution to the equation.

  20. Check It Out: Example 3B In most states, the price of each item is not the total cost. Sales tax must be added. If sales tax is 7.5%, the equation for total cost is c = 1.075p, where p is the price before tax. How much will a $10 item cost after sales tax? c = 1.075(10) The price of the item before tax is $10. Multiply. c = 10.75 After sales tax, the $10 item will cost $10.75, so (10, 10.75) is a solution to the equation.

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

  22. x 4x – 6 y (x, y)‏ 2 4(2) –6 2 (2, 2)‏ 4 4(4) –6 10 (4, 10)‏ 6 4(6) –6 18 (6, 18)‏ 8 4(8) –6 26 (8, 26)‏ 10 4(10) –6 34 (10, 34)‏ Lesson Quiz: Part I Determine whether each ordered pair is a solution of y = 4x –7. 1.(2, 15) 2. (4, 9)‏ 3. Use the given values to make a table of solutions. y = 4x6 for x = 2, 4, 6, 8, and 10 no yes

  23. Lesson Quiz: Part II 4. A plumbing company charges $50 for a service call and $15 per hour. They went on a 3-hour job. How much did the company earn? c = 15n + 50; $95

  24. Lesson Quiz for Student Response Systems 1. Which of the following ordered pairs is the solution of y = 6x – 9? A. (1, 3) B. (2, 3) C. (3, 2) D. (3, 7)

  25. Lesson Quiz for Student Response Systems 2. Which of the following ordered pairs is NOT the solution of y = 4x + 7? A. (4, 23) B. (5, 20) C. (1, 11) D. (3, 19)

  26. Lesson Quiz for Student Response Systems 3. Use the given values to identify the table of solutions. y = 6x – 4 for x = 1, 3, 5, 7, and 9 • B.

  27. Lesson Quiz for Student Response Systems 4. The water level in a tank is 17 feet. Due to the heavy rainfall, the water level increases at the rate of 7 feet per hour. What will be the water level in the tank after 5 hours? A.L = 17n + 17; 102 ft B.L = 17n + 7; 92 ft C.L = 7n + 17; 52 ft D.L = 7n + 7; 42 ft

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