A car service charges customers a flat fee per ride (which is higher during rush hour traffic) plus charges for each minute and each mile. Suppose that, in a certain metropolitan area during rush hour, the flat fee is $$ \$ 5 $$, the cost per minute is $$ \$ 0.30 $$, and the cost per mile is $$ \$ 1.20 $$. Let x be the number of minutes and y the number of miles. At the end of a ride, the driver said that the passenger owed $$ \$ 11.30 $$ and remarked that the number of minutes was three times the number of miles. Find the number of minutes and the number of miles for this trip. Complete the equation that represents the total cost of the ride. $$ \square=11.30 $$ (Do not include the $$ \$ $$ symbol in your answer. Do not simplify. Use integers or decimals for any numbers in the equation.)

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UpStudy AI · Accepted Answer

Given:
- Flat fee per ride during rush hour: $5
- Cost per minute: $0.30
- Cost per mile: $1.20
- Total cost of the ride: $11.30
- Number of minutes: 3 times the number of miles

Let's denote:
- Number of minutes: $$ x $$
- Number of miles: $$ y $$

The total cost of the ride can be represented by the equation:
$$ 5 + 0.30x + 1.20y = 11.30 $$

We are also given that the number of minutes is three times the number of miles, so we have the equation:
$$ x = 3y $$

Now, we need to solve these two equations to find the values of $$ x $$ and $$ y $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}5+0.3x+1.2y=11.3\x=3y\end{array}\right.$$
- step1: Substitute the value of $$x:$$
  $$5+0.3\times 3y+1.2y=11.3$$
- step2: Simplify:
  $$5+2.1y=11.3$$
- step3: Move the constant to the right side:
  $$2.1y=11.3-5$$
- step4: Subtract the numbers:
  $$2.1y=6.3$$
- step5: Divide both sides:
  $$\frac{2.1y}{2.1}=\frac{6.3}{2.1}$$
- step6: Divide the numbers:
  $$y=3$$
- step7: Substitute the value of $$y:$$
  $$x=3\times 3$$
- step8: Calculate:
  $$x=9$$
- step9: Calculate:
  $$\left\{ \begin{array}{l}x=9\y=3\end{array}\right.$$
- step10: Check the solution:
  $$\left\{ \begin{array}{l}x=9\y=3\end{array}\right.$$
- step11: Rewrite:
  $$\left(x,y\right) = \left(9,3\right)$$
The solution to the system of equations is $$ x = 9 $$ and $$ y = 3 $$.

Therefore, the number of minutes for this trip is 9 minutes, and the number of miles is 3 miles.

The equation that represents the total cost of the ride is:
$$ 5 + 0.30x + 1.20y = 11.30 $$

Substituting $$ x = 9 $$ and $$ y = 3 $$ into the equation, we get:
$$ 5 + 0.30(9) + 1.20(3) = 11.30 $$

Simplifying the equation, we have:
$$ 5 + 2.70 + 3.60 = 11.30 $$
$$ 11.30 = 11.30 $$

Therefore, the total cost of the ride is $11.30, which matches the given information.
The number of minutes is 9 and the number of miles is 3. The total cost equation is $$ 5 + 0.30x + 1.20y = 11.30 $$.

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