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# 7.2: Solving Linear Systems by Substitution

Difficulty Level: 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 1\begin{align*}1\end{align*}.

Encourage students to isolate the variable with a coefficient of 1\begin{align*}1\end{align*} or 1\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 x\begin{align*}x\end{align*} into one of the original equations in case that an error was made in solving for x\begin{align*}x\end{align*}.

General Tip: In fact, any of the two original equations can be used to find y\begin{align*}y\end{align*} once x\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 x=117.65\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 x\begin{align*}x\end{align*}.
• For problems where variables other than x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} are used, have students clearly state which variable is independent and which is dependent. See Review Questions 5-9.

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