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7.2: Solving Linear Systems by Substitution

Difficulty Level: At Grade Created by: CK-12
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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.


Terms introduced in this lesson:

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*}1.

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

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

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