# How do I use substitution when there is no pre solved variable?

5x + 5y = 20
3x + y = 23

#### Romsek

##### Junior Member
5x + 5y = 20
3x + y = 23
solve for one

$$\displaystyle 3x+y=23\\ y = 23-3x$$

#### Otis

##### Senior Member
How do I use substitution when there is no pre-solved variable?
You substitute an expression for a variable, instead of a number.

Solve one of the equations for x, and then substitute the resulting expression for x into the other equation. That gives a new equation containing only y.

Alternatively, you could solve one of the equations for y, and then substitute the resulting expression for y into the other equation. That gives a new equation containing only x.

Once you have a new equation containing only one variable, you can solve it. You'll then have a number to substitute into the formula that you found for the other variable.

EG:

x + y = 4
-2x + 2y = 1

I solve the first equation for y.

y = 4 - x

I now have a formula for finding y, if I know x.

Next, I substitute this expression for y into the second equation.

-2x + 2(4 - x) = 1

Now I have an equation containing only x. I solve it.

x = 7/4

To find y, I substitute 7/4 for x into the formula for y.

y = 4 - 7/4

y = 9/4

The solution to this example system of two equations is (7/4, 9/4).
:cool: