1. 7.2 Solving Linear Systems by Substitution

2. Steps: 1. Solve one of the equations for one of the variables. 2.Substitute that expression into the other equation and solve for the other variable. This gives you the first part of your ordered pair. 3.Substitute this value into the revised first equation and solve. This gives you the second part of your ordered pair. 4. 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 1.Rewrite: -x + y = 1 y = x + 1 (you could write x = y-1) 2.Substitute:2x + y = -2 2x + (x + 1) = -2 2x + x + 1 = -2 3x + 1 = -2 3x = -3 x = -1

4. 3.Substitute: y = x + 1 y = -1 + 1 y = 0 Solution (-1, 0) 4.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 x = -y + 1 Now where you have an x in the first equation, substitute in –y +1 and solve for y. 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 2x + 3 y = 5035 x + y = 1957 rewrite x = 1957 – y Substitute into 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