1. 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
Solve 1 equation for 1 variable. So rewrite:
-x + y = 1
+x x
y = x (or 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
-(-1) + 0 = 1
1 = 1
2x + y = -2
2(-1) + 0 = -2
-2 = -2
Both are true so the solution is (-1,0). Graph to check.

5. Can graph to find the approximate solution
Example:
2x = 5
x + y = 1
Can graph to find the approximate solution
Can you tell what the solution is?
Graphs do not provide precise solutions.
Now solve using substitution.

6. 1. Solve 1 equation for 1 variable.
2x = 5
x + y = 1
x + y = 1
-y -y
x = -y + 1
2x = 5
2(-y + 1) = 5
-2y + 2 = 5
-2y = 3
y = -3/2
2. Now substitute –y +1 into the first equation as x and solve for y.
3. Now substitute -3/2 into the second equation as y and solve for x.
x + y = 1
x + (-3/2) = 1
x = 5/2 or 2.5
Solution (5/2, -3/2)
Check ordered pair in 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 (one is typically a total amount of something, the other is often a price or value of the items – make sure to keep them separate to ensure you write the correct coefficients into the correct equations)
A + C = this equation represent the number of people, so A stands for number of adults, C is number of children, and set it = to total number of people
11.95A C = this equation represents cost so put the cost coefficient on each variable and have it = the total cost

8. A+C=6 rewrite as A=6-C (solve one equation for one variable)
11.95 ( 6 - C) C = (substitute that into A in the other equation)
71.70 – C C = (solve for C)
– 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
= 61.70
61.70 = (Check)