2. The Substitution Method
A method to solve a system of linear equations in 2 variables

3. When you “Solve a system of equations” you are looking for a solution that will solve every equation in the system (group).

4. ??? A linear equation Ax + By = C has an infinite number of solution? ???

5. How do we start with two equations , each having an infinite number solutions, and find the common solution (if any)???

6. This Method will create one combined equation with only one variable
This Method will create one combined equation with only one variable. This is the kind of equation that you can solve!

7. Solve one equation for one of the variables
Substitution Method - Step One
Solve one equation for one of the variables
Choose either equation and solve for either variable.
(Choose the easiest one the solve. )

8. Step One - Solve one equation for one of the variables
2x + y = 53 1
x + 5y = 139 2
Choose either equation and solve for either variable.
(Choose the easiest one the solve. )

9. 2x + y = 53 y = -2x +53 x + 5y = 139 x = - 5y +139 -2x -5y
Step One - Solve one equation for one of the variables
-2x
In this problem you could have solved equation #1 for y or solved equation # 2 for x.
2x + y = 53 1
y = -2x +53
-5y
x + 5y = 139 2
x = - 5y +139

10. Substitute this expression in the other equation and solve.
Step One - Solve one equation for one of the variables
Substitution Method - Step Two
Substitute this expression in the other equation and solve.

11. 2x + y = 53 y = -2x +53 x + 5y = 139 x + 5 (-2x +53) = 139
Substitute this expression in the other equation and solve.
If you solve for y in the first equation take this expression and substitute it in for y in the 2nd equation
2x + y = 53 1
y = -2x +53
x + 5y = 139 2
x + 5 (-2x +53) = 139
joined

12. This will create one combined equation with only one variable
This will create one combined equation with only one variable. This is the kind of equation that you can solve!

13. x + 5 (-2x +53) = 139 x + -10x+265 = 139 -9x + 265 = 139 -9x = -126
Substitute this expression in the other equation and solve.
2x + y = 53
y = -2x +53
Now solve for x 1
x + 5y = 139 2
x + 5 (-2x +53) = 139
joined
x + -10x+265 = 139
-9x = 139
-9x = -126
x = -126/-9=14

14. Find the corresponding value of the other variable.
After solving the combined equation ….
Find the corresponding value of the other variable.
(Substitute the value you found in step 2 to back into the equation)

15. y = -2x +53 y = -2(14)+53 (x , y ) = (14,25) y = -28+53=25 y = -2x +53
Substitute the value you found for the first variable back into one of the original equations
2x + y = 53 1
x + 5y = 139 2
x + 5 (-2x +53) = 139
x + -10x+265 = 139
-9x = 139
-9x = -126
x = -126/-9 = 14
From the last step you found that x was 14.
Take this value and plug it back into one of the original equations and find y.
y = -2x +53
y = -2(14)+53
(x , y ) = (14,25)
y = =25

16. 3 ( ) + 2y = 113 3 (4y +5) + 2y = 113 12y +15+2y = 113 14y + 15 = 113
Solve one of the equations for one of the variables
x = 4y +5
x - 4y = 5 1
3x + 2y = 113 2
Substitute this expression into the other equation
3 ( ) + 2y = 113
3 (4y +5) + 2y = 113
12y +15+2y = 113
14y + 15 = 113
14y = 98
y = 98/14 = 7
and solve.
Substitute the value you found for the first variable back into one of the original equations
x = 4y +5
x = 4(7) +5
x = 28+5=33
(x , y ) = (33,7)

17. Step One - Solve one equation for one of the variables
Step Two - Substitute this expression in the other equation and solve.
Step Three - Find the other variable (Substitute value back into one of the equations)

19. I’m thinking of two numbers
I’m thinking of two numbers. One number is one less then twice the other. The difference of the numbers is 18. Find the numbers.

20. I’m thinking of two numbers
I’m thinking of two numbers. One number is one less then twice the other. The difference of the numbers is 18. Find the numbers.
Let x and y represent the numbers. Write two equations to represent the relationships.
y-x=18
y=2x-1

21. x-1 = 18 x = 19 (x , y ) = (19,37) y = 2x - 1 y = 2x-1 y - x = 18
Solve one of the equations for one of the variables
y = 2x - 1
y = 2x-1 1
y - x = 18 2
Substitute this expression into the other equation
2x x = 18
x-1 = 18
x = 19
and solve.
Substitute the value you found for the first variable back into one of the original equations
y = 2x - 1
y = 2(19) - 1
x = 38-1=37
(x , y ) = (19,37)

23. John had all dimes and quarters worth $5
John had all dimes and quarters worth $5.45 If he had 35 coins in all, find out how many of each coin he had.

24. Let d = the # of dimes and q = the # of quarters Write two equations.
John had all dimes and quarters worth $5.45 If he had 35 coins in all, find out how many of each coin he had.
Let d = the # of dimes and q = the # of quarters Write two equations.
d+q=35
10d+25q=545

25. 10d + 25(35 - d) = 545 10d +875-25d = 545 -15d + 875 = 545 -15d = -330
Solve one of the equations for one of the variables
q = 35 - d
d + q = 35 1
10d + 25q = 545 2
Substitute this expression into the other equation
10d + 25(35 - d) = 545
10d d = 545
-15d = 545
-15d = -330
d = -330/-15 = 22
and solve.
q = 35 - d
q =
q = 13
Substitute the value you found for the first variable back into one of the original equations
# of dimes = 22
# of quarters =13