To find the number of miles at which both companies have the same service price, we need to solve the system of linear equations by graphing.

Given the equations:

• Company A: $y = 0.25x + 15$
• Company B: $y = 0.50x$

Step-by-step process:

1. Find the point of intersection by setting both equations equal to each other:

$0.25x + 15 = 0.50x$

2. Solve for $x$:

Subtract $0.25x$ from both sides:

$15 = 0.50x - 0.25x$

$15 = 0.25x$

Now, divide both sides by 0.25:

$x = \frac{15}{0.25} = 60$

So, the two companies will charge the same price after 60 miles.

3. Find the price by plugging $x = 60$ into either equation. We'll use Company B's equation for simplicity:

$y = 0.50(60) = 30$

So, the price for both companies at 60 miles will be $30.

Conclusion:

• The companies will charge the same price at 60 miles.
• The price at this point is $30.

Would you like more details, or do you have other questions? Here are some related questions you can explore:

1. How does the initial fee in Company A's pricing affect the overall cost for different distances?
2. How would the costs change if the initial fee for Company A were different?
3. What happens if Company B has a different per-mile charge, say $0.75 per mile?
4. How would the equations change if both companies introduced a discount for longer rentals?
5. How can you use the concept of linear equations to compare other types of services with variable costs?

Tip: When solving systems of equations by graphing, plotting the two equations on the same graph can visually show the point of intersection, which is the solution to the system.