5.1 Solve systems of equations by graphing  (Page 5/14)

Solution

$\begin{array}{cccc}\begin{array}{c}\text{We will compare the slope and intercepts of the two lines.}\hfill \\ \\ \\ \end{array}\hfill & & & \{\begin{array}{ccc}\hfill 2x+y& =\hfill & \mathrm{-3}\hfill \\ \hfill x-5y& =\hfill & 5\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{Write both equations in slope–intercept form.}\hfill \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array}\hfill & & & \hfill \begin{array}{cccccccc}\hfill 2x+y& =\hfill & \mathrm{-3}\hfill & & & \hfill x-5y& =\hfill & 5\hfill \\ \hfill y& =\hfill & \mathrm{-2}x-3\hfill & & & \hfill \mathrm{-5}y& =\hfill & \text{−}x+5\hfill \\ & & & & & \hfill \frac{\mathrm{-5}y}{\mathrm{-5}}& =\hfill & \frac{\text{−}x+5}{\mathrm{-5}}\hfill \\ & & & & & \hfill y& =\hfill & \frac{1}{5}x-1\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{Find the slope and intercept of each line.}\hfill \\ \\ \\ \\ \\ \\ \end{array}\hfill & & & \hfill \begin{array}{cccccccccc}\hfill y& =\hfill & \mathrm{-2}x-3\hfill & & & & & \hfill y& =\hfill & \frac{1}{5}x-1\hfill \\ \hfill m& =\hfill & \mathrm{-2}\hfill & & & & & \hfill m& =\hfill & \frac{1}{5}\hfill \\ \hfill b& =\hfill & \mathrm{-3}\hfill & & & & & \hfill b& =\hfill & \mathrm{-1}\hfill \end{array}\hfill \\ & & & \text{Since the slopes are different, the lines intersect.}\hfill \end{array}$

A system of equations whose graphs are intersect has 1 solution and is consistent and independent.

Solution

$\begin{array}{cccc}\begin{array}{c}\text{We will compare the slope and intercepts of the two lines.}\hfill \\ \\ \\ \\ \end{array}\hfill & & & \hfill \{\begin{array}{ccc}\hfill 3x-2y& =\hfill & 4\hfill \\ \hfill y& =\hfill & \frac{3}{2}x-2\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{Write the first equation in slope–intercept form.}\hfill \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array}\hfill & & & \hfill \begin{array}{ccc}\hfill 3x-2y& =\hfill & 4\hfill \\ \hfill \mathrm{-2}y& =\hfill & \mathrm{-3}x+4\hfill \\ \hfill \frac{\mathrm{-2}y}{\mathrm{-2}}& =\hfill & \frac{\mathrm{-3}x+4}{\mathrm{-2}}\hfill \\ \hfill y& =\hfill & \frac{3}{2}x-2\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{The second equation is already in}\hfill \\ \text{slope–intercept form.}\hfill \end{array}\hfill & & & \hfill \begin{array}{ccc}\hfill y& =\hfill & \frac{3}{2}x-2\hfill \end{array}\hfill \\ & & & \begin{array}{c}\text{Since the equations are the same, they have the same slope}\hfill \\ \text{and same}y\text{-intercept and so the lines are coincident.}\hfill \end{array}\hfill \end{array}$

A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.

Solution

Step 1. Read the problem.

Step 2. Identify what we are looking for.

We are looking for the number of quarts of fruit juice and the number of quarts of club soda that Sondra will need.

Step 3. Name what we are looking for. Choose variables to represent those quantities.

  Let $f=$ number of quarts of fruit juice.
     $c=$ number of quarts of club soda

Step 4. Translate into a system of equations.

We now have the system. $\{\begin{array}{c}f+c=10\hfill \\ f=4c\hfill \end{array}$

Step 5. Solve the system of equations using good algebra techniques.

The point of intersection (2, 8) is the solution. This means Sondra needs 2 quarts of club soda and 8 quarts of fruit juice.

Step 6. Check the answer in the problem and make sure it makes sense.

Does this make sense in the problem?

Yes, the number of quarts of fruit juice, 8 is 4 times the number of quarts of club soda, 2.

Yes, 10 quarts of punch is 8 quarts of fruit juice plus 2 quarts of club soda.

Step 7. Answer the question with a complete sentence.

Sondra needs 8 quarts of fruit juice and 2 quarts of soda.