An auto shop has two mechanics. They charge a combined rate of  210 per hour. On their most recent car, the first mechanic worked for 10 hours, and the second mechanic worked for 5 hours. Together they charged a total of 1625 . What is the rate charged per hour by each mechanic?

To determine the hourly rate charged by each mechanic, we need to set up a system of equations based on the given information.

Let's denote:
- \( r_1 \) as the hourly rate of the first mechanic.
- \( r_2 \) as the hourly rate of the second mechanic.

From the problem statement, we know two key pieces of information:
1. The combined rate of both mechanics per hour is \( 210 \) dollars.
2. The total charge for their work on the most recent car was \( 1625 \) dollars, where the first mechanic worked for 10 hours, and the second mechanic worked for 5 hours.

We can translate these pieces of information into the following system of equations:

1. \( r_1 + r_2 = 210 \)
2. \( 10r_1 + 5r_2 = 1625 \)

To solve this system, we can use the substitution or elimination method. Here, we will use the substitution method.

First, solve the first equation for \( r_1 \):
\[ r_1 = 210 - r_2 \]

Next, substitute this expression for \( r_1 \) into the second equation:
\[ 10(210 - r_2) + 5r_2 = 1625 \]

Distribute the 10:
\[ 2100 - 10r_2 + 5r_2 = 1625 \]

Combine like terms:
\[ 2100 - 5r_2 = 1625 \]

Subtract 1625 from both sides:
\[ 475 = 5r_2 \]

Solve for \( r_2 \):
\[ r_2 = \frac{475}{5} = 95 \]

Now that we have \( r_2 \), substitute it back into the expression for \( r_1 \):
\[ r_1 = 210 - 95 = 115 \]

Thus, the hourly rate for the first mechanic is \( 115 \) dollars, and the hourly rate for the second mechanic is \( 95 \) dollars.

To summarize:
- The rate charged by the first mechanic (\( r_1 \)) is \( \boxed{115} \) dollars per hour.
- The rate charged by the second mechanic (\( r_2 \)) is \( \boxed{95} \) dollars per hour.

Would you like more details on any step or have any questions?

Here are 8 related questions you might consider:
1. How would the solution change if the total charge was different?
2. What if the combined hourly rate of the mechanics was increased or decreased?
3. How do we handle situations where more than two variables are involved?
4. Can these methods be applied to non-linear equations?
5. How would the solution change if the number of hours worked by each mechanic was different?
6. Can you explain the elimination method for solving systems of equations?
7. What are some real-life applications of systems of linear equations?
8. How can graphing be used to solve a system of linear equations?

**Tip:** When solving systems of equations, always verify your solution by substituting the values back into the original equations to ensure they satisfy all conditions.

Learn how to solve a system of linear equations to determine hourly rates for mechanics. This problem involves using the substitution method to find rates based on total charges and hours worked. Suitable for high school students.

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