5.4 Solve applications with systems of equations  (Page 3/11)

Solution

$\begin{array}{cccc}\mathbf{\text{Step 1. Read}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\text{what we are looking for.}\hfill & & \text{We are looking for the measure of each angle.}\hfill \\ \\ \\ \begin{array}{c}\mathbf{\text{Step 3. Name}}\text{what we are looking for.}\hfill \\ \\ \\ \\ \end{array}\hfill & & \begin{array}{c}\text{Let}x=\text{the measure of the first angle.}\hfill \\ y=\text{the measure of the second angle}\hfill \end{array}\hfill \\ \\ \\ \mathbf{\text{Step 4. Translate}}\text{into a system of equations.}\hfill & & \text{The angles are complementary.}\hfill \\ & & \hfill x+y=90\hfill \\ & & \text{The difference of the two angles is 26 degrees.}\hfill \\ & & \hfill x-y=26\hfill \\ \\ \\ \text{The system is}\hfill & & \hfill \{\begin{array}{c}x+y=90\hfill \\ x-y=26\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\mathbf{\text{Step 5. Solve}}\text{the system of equations by elimination.}\hfill \\ \\ \\ \\ \end{array}\hfill & & \hfill \begin{array}{c}\underset{\text{\_\_\_\_\_\_\_\_\_}}{\{\begin{array}{l}x+y=90\hfill \\ x-y=26\hfill \end{array}}\hfill \\ 2x=116\hfill \end{array}\hfill \\ & & \hfill x=58\hfill \\ \begin{array}{}\\ \text{Substitute}x=58\text{into the first equation.}\hfill \end{array}\hfill & & \hfill \begin{array}{ccc}\hfill x+y& =\hfill & 90\hfill \\ \hfill 58+y& =\hfill & 90\hfill \\ \hfill y& =\hfill & 32\hfill \end{array}\hfill \\ \\ \\ \mathbf{\text{Step 6. Check}}\text{the answer in the problem.}\hfill & & \\ \begin{array}{c}\hfill 58+42=90✓\hfill \\ \hfill 58-32=26✓\hfill \end{array}\hfill \\ \\ \\ \mathbf{\text{Step 7. Answer}}\text{the question.}\hfill & & \text{The angle measures are 58 degrees and 42 degrees.}\hfill \end{array}$

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	We are looking for the measure of each angle.
Step 3. Name what we are looking for.	Let $x=$ the measure of the first angle. $y=$ the measure of the second angle
Step 4. Translate into a system of equations.	The angles are supplementary.
	The larger angle is twelve less than five times the smaller angle
The system is: Step 5. Solve the system of equations substitution.
Substitute 5 x − 12 for y in the first equation.
Solve for x .
Substitute 32 for in the second equation, then solve for y .
Step 6. Check the answer in the problem. $\begin{array}{ccc}\hfill 32+158& =\hfill & 180✓\hfill \\ \hfill 5·32-12& =\hfill & 147✓\hfill \end{array}$
Step 7. Answer the question.	The angle measures are 148 and 32.