Write a system of equations to describe the situation below, solve using elimination, and fill in
the blanks.
To keep in shape, Miranda exercises at a track near her home. She requires 26 minutes to do
6 laps running and 5 laps walking. In contrast, she requires 34 minutes to do 10 laps running
and 6 laps walking. Assuming she maintains a consistent pace while running and while
walking, how long does Miranda take to complete a lap?
Miranda takes
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UpStudy AI · Accepted Answer

To solve this problem, we can set up a system of equations based on the information provided. Let's denote the time it takes Miranda to complete one lap of running as $$ r $$ minutes and the time it takes her to complete one lap of walking as $$ w $$ minutes.

From the given information, we can form the following equations:

1. For 6 laps running and 5 laps walking:
$$ 6r + 5w = 26 $$

2. For 10 laps running and 6 laps walking:
$$ 10r + 6w = 34 $$

Now, we can solve this system of equations using the elimination method to find the values of $$ r $$ and $$ w $$.

Let's proceed with the elimination method to solve the system of equations.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}6r+5w=26\10r+6w=34\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}r=\frac{26-5w}{6}\10r+6w=34\end{array}\right.$$
- step2: Substitute the value of $$r:$$
  $$10\times \frac{26-5w}{6}+6w=34$$
- step3: Multiply the terms:
  $$\frac{5\left(26-5w\right)}{3}+6w=34$$
- step4: Multiply both sides of the equation by LCD:
  $$\left(\frac{5\left(26-5w\right)}{3}+6w\right)\times 3=34\times 3$$
- step5: Simplify the equation:
  $$130-7w=102$$
- step6: Move the constant to the right side:
  $$-7w=102-130$$
- step7: Subtract the numbers:
  $$-7w=-28$$
- step8: Change the signs:
  $$7w=28$$
- step9: Divide both sides:
  $$\frac{7w}{7}=\frac{28}{7}$$
- step10: Divide the numbers:
  $$w=4$$
- step11: Substitute the value of $$w:$$
  $$r=\frac{26-5\times 4}{6}$$
- step12: Simplify:
  $$r=1$$
- step13: Calculate:
  $$\left\{ \begin{array}{l}r=1\w=4\end{array}\right.$$
- step14: Check the solution:
  $$\left\{ \begin{array}{l}r=1\w=4\end{array}\right.$$
- step15: Rewrite:
  $$\left(r,w\right) = \left(1,4\right)$$
The solution to the system of equations is $$ r = 1 $$ and $$ w = 4 $$.

This means that Miranda takes 1 minute to complete one lap of running and 4 minutes to complete one lap of walking.

Therefore, Miranda takes 1 minute to complete a lap.
Miranda takes 1 minute to complete a lap.

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