# Systems of Linear Equations

## Solving Systems of Equations by Substitution

Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.
In this section, we will solve systems of linear equations by the substitution method.

### Solve a System of Equations by Substitution

We will use the same system we used first for graphing.
We will first solve one of the equations for either x or y. We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.
Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!
After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.
We’ll fill in all these steps now in .
If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in .
If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y.
In  it was easiest to solve for y in the first equation because it had a coefficient of 1. In  it will be easier to solve for x.
When both equations are already solved for the same variable, it is easy to substitute!
Be very careful with the signs in the next example.
In , it will take a little more work to solve one equation for x or y.
Look back at the equations in . Is there any way to recognize that they are the same line?
Let’s see what happens in the next example.

### Solve Applications of Systems of Equations by Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.
Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?
In the , we’ll use the formula for the perimeter of a rectangle, P = 2L + 2W.
For  we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.

### Key Concepts

1. Solve a system of equations by substitution



### Practice Makes Perfect

Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve Applications of Systems of Equations by Substitution
In the following exercises, translate to a system of equations and solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?