Solve the following system of equations by substitution. Show all steps. f(x)=-x^2 + 2x+3
g(x)=-2x+3

graph can be used to show the solution for a system of two linear equations. However, accurately determining the solution from a graph is not always easy or accurate. For example, where do you think the two lines shown below intersect?

It looks like they might intersect at (1.8, –0.7)—though this is only an estimate. In cases like this, you can use algebraic methods to find exact answers. One method to look at is called the substitution method. You solve one equation for one variable and then substitute this expression into the other equation.

Using Substitution to Solve a System of Equations

In the substitution method you solve for one variable, and then substitute that expression into the other equation. The important thing here is that you are always substituting values that are equivalent.

For example:

Sean is 5 years older than four times his daughter’s age. His daughter is 7. How old is Sean?

You might do this problem in your head. Sean’s daughter is 7, so “four times his daughter’s age” is 28, and 5 years added to that is 33. Sean is 33.

If you solved the problem like that, you used a simple substitution—you substituted in the value “7” for “his daughter’s age.” You learned in the second part of the problem that “his daughter is 7.” So substituting in a value of “7” for “his daughter’s age” in the first part of the problem was okay, because you knew these two quantities were equal.