Let's extract and solve the problem step by step.

Problem Summary:

An auto shop has two mechanics with a combined rate of $175 per hour. The first mechanic worked for 10 hours, and the second mechanic worked for 15 hours on a recent car job. Together, they charged a total of $2150. We need to find the hourly rate charged by each mechanic.

Let’s Define the Variables:

• Let $x$ be the rate charged per hour by the first mechanic.
• Let $y$ be the rate charged per hour by the second mechanic.

Given Equations:

1. The combined rate per hour for both mechanics:
   $x + y = 175$

2. The total charge for the job is $2150, calculated from the hours worked by each mechanic:
   $10x + 15y = 2150$

Solving the System of Equations:

We have a system of two linear equations:

1. $x + y = 175$
2. $10x + 15y = 2150$

We can solve this system using substitution or elimination. Let's use the substitution method:

1. From the first equation, solve for $y$: $y = 175 - x$

2. Substitute $y$ in the second equation: $10x + 15(175 - x) = 2150$

Let's solve this equation step-by-step.The solution to the system of equations is:

• The first mechanic charges $95 per hour.
• The second mechanic charges $80 per hour.

Summary:

• Combined rate per hour: $x + y = 95 + 80 = 175$ (consistent with the given information).
• Total charge for the job: $10 \times 95 + 15 \times 80 = 950 + 1200 = 2150$ (also consistent with the given total).

If you have any further questions or need more details, feel free to ask!

Related Questions:

1. How would the solution change if the combined rate was different?
2. What if both mechanics worked the same number of hours?
3. How can we use matrices to solve this system of equations?
4. What would happen if the total charge was calculated with a different number of hours?
5. Can you explain how to graphically represent this system of equations?

Tip:

When solving a system of linear equations, double-check the given conditions to ensure the equations accurately represent the problem scenario.