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Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

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Section 3.6 Systems of Linear Equations (LF6)

Subsection 3.6.1 Activities

Observation 3.6.1.

Often times when solving a real-world application, more than one equation is necessary to describe the information. We’ll investigate some of those in this section.
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Activity 3.6.2.

Admission into a carnival for \(4\) children and \(2\) adults is \(\$128.50\text{.}\) For \(6\) children and \(4\) adults, the admission is \(\$208\text{.}\) Assuming a different price for children and adults, what is the price of the child’s admission and the price of the adult admission?
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(a)
Let \(c\) represent the cost of a child’s admission and \(a\) the cost of an adult admission. Write an equation to represent the total cost for \(4\) children and \(2\) adults.
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  1. \(\displaystyle 2c+4a=\$128.50\)
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  2. \(\displaystyle c+a=\$128.50\)
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  3. \(\displaystyle 4c+2a=\$128.50\)
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  4. \(\displaystyle c+a=\$336.50\)
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Answer.
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(b)
Now write an equation to represent the total cost for \(6\) children and \(4\) adults.
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  1. \(\displaystyle 6c+4a=\$208\)
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  2. \(\displaystyle c+a=\$208\)
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  3. \(\displaystyle 4c+6a=\$208\)
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  4. \(\displaystyle c+a=\$336.50\)
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Answer.
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(c)
Using the above equations, check by substitution which admission prices would satisfy both equations?
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  1. \(c=\$20\) and \(a=\$24.25\)
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  2. \(c=\$24.50\) and \(a=\$15.25\)
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  3. \(c=\$20\) and \(a=\$22\)
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  4. \(c=\$14.50\) and \(a=\$30.25\)
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Answer.
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Definition 3.6.3.

A system of linear equations consists of two or more linear equations made up of two or more variables. A solution to a system of equations is a value for each of the variables that satisfies all the equations at the same time.
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Activity 3.6.4.

Consider the following system of equations.
\begin{equation*} \begin{cases} y=2x+4\\ 3x+2y=1 \end{cases} \end{equation*}
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Which of the ordered pairs is a solution to the system?
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  1. \(\displaystyle (3,10)\)
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  2. \(\displaystyle (0,4)\)
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  3. \(\displaystyle (1,-1)\)
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  4. \(\displaystyle (-1,2)\)
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Answer.
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Remark 3.6.5.

While we can test points to determine if they are solutions, it is not feasible to test every possible point to find a solution. We need a method to solve a system.
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Activity 3.6.6.

Consider the following system of equations.
\begin{equation*} \begin{cases} 3x-y=2\\ x+4y=5 \end{cases} \end{equation*}
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(a)
Rewrite the first equation in terms of \(y\text{.}\)
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Answer.
\(y=3x-2\)
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(b)
Rewrite the second equation in terms of \(y\text{.}\)
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Answer.
\(y=\dfrac{-x+5}{4}\)
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(c)
Graph the two equations on the same set of axes. Where do the lines intersect?
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Answer.
\((1,1)\text{.}\)
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(d)
Check that the point of intersection of the two lines is a solution to the system of equations.
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Answer.
To check to see if the point of intersection is a solution, plug in \(1\) in for \(x\) and \(1\) in for \(y\) into BOTH equations.
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Remark 3.6.7.

Sometimes it is difficult to determine the exact intersection point of two lines using a graph. Let’s explore another possible method for solving a system of equations.
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Activity 3.6.8.

Consider the following system of equations.
\begin{equation*} \begin{cases} 3x+y=4\\ x+3y=10 \end{cases} \end{equation*}
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(a)
Graph the two equations on the same set of axes. Is it possible to determine exactly where the lines intersect?
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Answer.
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(b)
Solve the first equation for \(y\) and substitute into the second equation. What is the resulting equation?
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  1. \(\displaystyle x+4-3x=10\)
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  2. \(\displaystyle x+3(4-3x)=10\)
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  3. \(\displaystyle 4-3x+3y=10\)
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  4. \(\displaystyle 3x+(4-3x)=4\)
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Answer.
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(d)
Substitute the value of \(x\) into the first equation to find the value of \(y\text{.}\)
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  1. \(\displaystyle y=-5\)
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  2. \(\displaystyle y=\dfrac{13}{4}\)
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  3. \(\displaystyle y=-3\)
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  4. \(\displaystyle y=\dfrac{4}{3}\)
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Answer.
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(e)
Write the solution to the system of equations (the found values of \(x\) and \(y\)) as an ordered pair.
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Answer.
\((0.25, 3.25)\)
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Remark 3.6.9.

This method of solving a system of equations is referred to as the Substitution Method.
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  1. Solve one of the equations for one variable.
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  2. Substitute the expression into the other equation to solve for the remaining variable.
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  3. Substitute that value into either equation to find the value of the first variable.
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Activity 3.6.10.

Solve the following system of equations using the substitution method.
\begin{equation*} \begin{cases} x+2y=-1\\ -x+y=3 \end{cases} \end{equation*}
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Answer.
\(\left( -\dfrac{7}{3}, \dfrac{2}{3} \right)\)
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Remark 3.6.11.

While the substitution method will always work, sometimes the resulting equations will be difficult to solve. Let’s explore a third method for solving a system of two linear equations with two variables.
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Activity 3.6.12.

Consider the following system of equations.
\begin{equation*} \begin{cases} 5x+7y=12\\ 3x-7y=37 \end{cases} \end{equation*}
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(b)
Use the resulting equation after addition, to solve for the variable.
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  1. \(\displaystyle x=-\dfrac{15}{2}\)
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  2. \(\displaystyle y=\dfrac{49}{14}\)
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  3. \(\displaystyle x=\dfrac{49}{22}\)
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  4. \(\displaystyle x=\dfrac{49}{8}\)
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Answer.
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(c)
Use the value to find the solution to the system of equations.
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Answer.
\(\left( \dfrac{49}{8}, -\dfrac{149}{56}\right)\)
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Remark 3.6.13.

This method of solving a system of equations is referred to as the Elimination Method.
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  1. Combine the two equations using addition or subtraction to eliminate one of the variables.
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  2. Solve the resulting equation.
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  3. Substitute that value into either equation to find the value of the other variable.
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Activity 3.6.14.

Solve the following system of equations using the elimination method.
\begin{equation*} \begin{cases} 7x-4y=3\\ 3y-7x=8 \end{cases} \end{equation*}
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Answer.
\(\left(-\dfrac{41}{7} ,-11 \right)\)
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Activity 3.6.15.

Consider the following system of equations.
\begin{equation*} \begin{cases} 5x-9y=6\\ -10x+4y=2 \end{cases} \end{equation*}
Notice that if you simply add the two equations together, it will not eliminate a variable. Substitution will also be difficult since it involves fractions.
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(a)
What value can you multiply the first equation by so that when you add the result to the second equation one variable cancels?
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  1. \(-1\) and the \(x\) will cancel
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  2. \(2\) and the \(x\) will cancel
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  3. \(3\) and the \(y\) will cancel
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  4. \(-2\) and the \(y\) will cancel
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Answer.
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Activity 3.6.16.

For each system of equations, determine which method (graphical, substitution, or elimination) might be best for solving.
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Subsection 3.6.2 Exercises

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