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From the images you uploaded, the problem you're working on involves comparing costs for renting a moving van from two companies, Company A and Company B. Let's break it down step by step and solve it.

### Problem Breakdown:
1. **Company A**: Charges $39.95 per day plus $0.79 per mile.
2. **Company B**: Charges $55.95 per day plus $0.29 per mile.

You're asked to do the following:

#### Part a:
**Write two equations** representing the costs of renting a van from each company:
- Let $ A $ represent the total daily rental cost from Company A and $ B $ represent the total daily rental cost from Company B.
- Let $ m $ represent the number of miles driven in one day.

The equations for each company's costs would be:

- **Company A**: $ A(m) = 39.95 + 0.79m $
- **Company B**: $ B(m) = 55.95 + 0.29m $

These equations show the daily costs for each company as a function of miles driven $ m $.

#### Part b:
**Evaluate the first equation if $ m = 37 $:**

For Company A:
\[
A(37) = 39.95 + 0.79(37)
\]

Let's calculate that:
\[
A(37) = 39.95 + 29.23 = 69.18
\]

So, the total cost for Company A if you drive 37 miles is **$69.18**.

#### Part c:
**Evaluate the second equation if $ m = 42 $:**

For Company B:
\[
B(42) = 55.95 + 0.29(42)
\]

Let's calculate that:
\[
B(42) = 55.95 + 12.18 = 68.13
\]

So, the total cost for Company B if you drive 42 miles is **$68.13**.

#### Part d:
**Set up a new equation to determine when the costs of renting a van from Company A will be equal to the cost of renting a van from Company B**:
We set the two equations equal to each other:
\[
39.95 + 0.79m = 55.95 + 0.29m
\]

#### Part e:
**Rewrite the equation from part d and solve for $ m $:**

Subtract $ 0.29m $ from both sides:
\[
39.95 + 0.50m = 55.95
\]

Subtract 39.95 from both sides:
\[
0.50m = 16
\]

Solve for $ m $:
\[
m = \frac{16}{0.50} = 32
\]

So, the costs will be equal when $ m = 32 $ miles.

#### Part f:
**State the answer in a complete sentence**:
The cost of renting a van from Company A will be equal to the cost of renting a van from Company B when you drive 32 miles in a day.

---

Do you need further details or have any questions about these steps?

### Relative Questions:
1. What happens if you drive fewer than 32 miles? Which company will be cheaper?
2. How does changing the daily rental cost affect the breakeven point?
3. What would the breakeven point be if Company A charged $1.00 per mile instead of $0.79?
4. Can you graph both cost equations and observe their intersection at 32 miles?
5. How do increases in gas prices influence which company is the better deal over longer distances?

### Tip:
When solving equations that compare costs, always isolate the variable you're solving for (in this case, miles) by eliminating smaller terms first!

You rented a moving van from Company A and Company B. Company A charges $39.95 per day plus $0.79 per mile, while Company B charges $55.95 per day plus $0.29 per mile. Write equations for the total costs of each company, and determine when the cost of Company A equals the cost of Company B.

Learn how to calculate and compare the cost of renting a moving van from two companies. Company A charges $39.95 per day plus $0.79 per mile, while Company B charges $55.95 per day plus $0.29 per mile. This step-by-step guide includes equations, calculations, and solutions to find when the costs are equal.

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