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7.2 Solving Linear Systems by Substitution

7.2 Solving Linear Systems by Substitution

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7.2 Solving Linear Systems by Substitution

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  1. 7.2 Solving Linear Systems by Substitution

  2. Steps: 1. Solve one of the equations for one of the variables. • Substitute that expression into the other equation and solve for the other variable. This gives you the first part of your ordered pair. • Substitute this value into the revised first equation and solve. This gives you the second part of your ordered pair. • Check the solution pair in each of the original equations. If it works, you have the solution.

  3. Example: -x + y = 1 2x + y = -2 • Rewrite: -x + y = 1 y = x + 1 (you could write x = y-1) • Substitute: 2x + y = -2 2x + (x + 1) = -2 2x + x + 1 = -2 3x + 1 = -2 3x = -3 x = -1

  4. Substitute: y = x + 1 y = -1 + 1 y = 0 Solution (-1, 0) • Check: -x + y = 1 2x + y = -2 -(-1) + 0 = 1 2(-1) + 0 = -2 1 = 1 -2 = -2 Both are true so the solution is (-1,0). Graph to check.

  5. Graph to find the solution • 2x = 5 • x + y = 1 Can you tell what the solution is? Now solve using substitution.

  6. Change the second equation by solving for x Now where you have an x in the first equation, substitute in –y +1 and solve for y. x = -y + 1 You get y = -3/2 Now plug -3/2 into the second equation and solve for x. You find that x = 5/2. Solution (5/2, -3/2) Check each equation to make sure you have the right answer.

  7. Real world example: Dinner at a China Buffet Adult cost is $11.95 Children cost $6.95 Total bill is $61.70 Total number of people is 6 How many adults and how many children went? Write 2 equations 11.95A + 6.95C = 61.70 A + C = 6 (rewrite A = 6 – C) Substitute

  8. 11.95 ( 6 - C) + 6.95 C = 61.70 (substitute) 71.70 – 11.95 C + 6.95 C = 61.70 71.70 – 5 C = 61.70 - 5C = -10 C = 2 A + C = 6 A + 2 = 6 A = 4 There are 4 adults and 2 children. 4(11.95) + 2(6.95) = 61.70 47.80 + 13.90 = 61.70 61.70 = 61.70 (Check)

  9. Real world example: (#30 ) Tickets Sold Student price $2 General Admission $3 Total amount collected $5035 Total number of tickets sold is 1957 How many adults and how many children went? Write 2 equations x + y = 1957 rewrite x = 1957 – y Substitute into 2x + 3 y = 5035 2x + 3 y = 5035

  10. 2(1957 – y) + 3y = 5035 3914 – 2y + 3y = 5035 3914 + y = 5035 -3914 = --3914 y = 1121 x = 1957 – y x = 1957-1121 X = 836 836 student tickets & 1121 general admission tickets sold. 2(236) + 3(1121) = 5035 1672 + 3363 = 5035 5035 = 5035