1. Solving By Substitution

2. Linear systems can be solved algebraically.
One way is to use substitution.
This is easiest when one or both equations is given in terms of either x or y.
This means that the x or y is alone on one side of the equal sign.

3. Examples Both equations given in terms of y y = 3x + 2 y = -4x - 7
One in terms of y, one in terms of x
y = x x = 3y - 2
One equation in terms of y
y = 2x 3y = x - 1
One equation in terms of x
x = 3y -2x = y - 3

4. Solving With Substitution
Solve the linear system:
y = 2x + 1
y = 3x - 2

5. Substitute the value of y in equation 1 into equation 2.
Step #1
Substitute the value of y in equation 1 into equation 2.
y = 3x – 2  equation 2
y = 2x + 1 equation 1

6. Substitute the value of y in equation 1 into equation 2.
Step #1
Substitute the value of y in equation 1 into equation 2.
2x + 1 = 3x – 2

7. Step #2 Solve for x: 2x + 1 = 3x - 2 Subtract 2x from both sides.
2x – 2x + 1 = 3x – 2x – 2
1 = x – 2
Add 2 to both sides ..
1 + 2 = x – 2 + 2
3 = x

8. Substitute the value of x we just found into either equation 1 or 2
Step #3
Substitute the value of x we just found into either equation 1 or 2
y = 2x + 1
y = 2(3) + 1
y = 6 + 1
y = 7

9. Final Answer Write the final answer as a coordinate point.
So, we found our answer to be
x = 3 and y = 7.
So our solution is (3, 7)

10. Remember
The solution to a linear system is the point of intersection of two equations.
That is where the two lines cross and share the same value for x and y.