# 7.2: Solving Linear Systems by Substitution

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Solve systems of equations in two variables by substituting for either variable.
- Manipulate
*standard form*equations to isolate a single variable. - Solve real-world problems using systems of equations.
- Solve mixture problems using systems of equations.

## Vocabulary

Terms introduced in this lesson:

- substitution
- substitution method
- standard form of a linear equation

## Teaching Strategies and Tips

Use Example 2 to motivate the substitution method.

- As the solution consists of fractions, the system is more complicated than any example presented up to this point.
- Emphasize that the graphing method can only provide an approximation. Therefore a different method is needed.

The substitution method:

- An algebraic method; provides exact solutions.
- A technique for replacing an unknown with another expression to obtain a third equation with only one unknown (replacing
*equals with equals*). - Best used when one of the coefficients of the variables is \begin{align*}1\end{align*}.

Encourage students to isolate the variable with a coefficient of \begin{align*}1\end{align*} or \begin{align*}-1\end{align*}. Students often give themselves extra work by choosing an equation and a variable at random.

Mixture problems:

- Mixtures do not necessarily pertain to chemistry. See Example 4 and
*Review Question*7. - Approach Example 7 with a picture. By the labeling the unknowns in it, the system of equations will be evident.

## Error Troubleshooting

In Example 2, have students back-substitute \begin{align*}x\end{align*} into one of the *original* equations in case that an error was made in solving for \begin{align*}x\end{align*}.

General Tip: In fact, *any* of the two original equations can be used to find \begin{align*}y\end{align*} once \begin{align*}x\end{align*} is determined. The easier-looking the equation the better.

Point out in Example 3 that the question can be answered after determining \begin{align*}x=117.65\end{align*}. There is no need to back-substitute to find the *cost per month.*

General Tip: Remind students to write coordinates in correct order, depending on how they labeled the variables in the beginning.

- Students often incorrectly write the value they found first as \begin{align*}x\end{align*}.
- For problems where variables other than \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are used, have students clearly state which variable is independent and which is dependent. See
*Review Questions*5-9.