Question

One month Dan rented 12 movies and 2 video games for a total of $31. The next month he rented 3 movies and 5 video games for a total of $37. Find the rental cost for each movie and each video game.'

Answer 1

Let's denote the rental cost of each movie as \( m \) and the rental cost of each video game as \( v \).

We can set up the following system of equations based on the information provided:

1. For the first month:
\[
12m + 2v = 31
\]

2. For the second month:
\[
3m + 5v = 37
\]

Now we can solve this system of equations.

**Step 1:** Simplify the first equation.
We can divide the first equation by 2 to make it simpler:
\[
6m + v = 15.5 \quad \text{(Equation 1)}
\]

**Step 2:** Rearrange Equation 1 to express \( v \):
\[
v = 15.5 - 6m \quad \text{(Equation 2)}
\]

**Step 3:** Substitute Equation 2 into the second equation:
Substituting \( v \) in the second equation:
\[
3m + 5(15.5 - 6m) = 37
\]
Expand this:
\[
3m + 77.5 - 30m = 37
\]
Combine like terms:
\[
-27m + 77.5 = 37
\]
Subtract 77.5 from both sides:
\[
-27m = 37 - 77.5
\]
\[
-27m = -40.5
\]

**Step 4:** Solve for \( m \):
\[
m = \frac{-40.5}{-27} = \frac{40.5}{27} = \frac{135}{90} = \frac{15}{10} = 1.5
\]
So, \( m = 1.5 \).

**Step 5:** Find \( v \) using Equation 2:
Substituting \( m = 1.5 \) back into \( v \) equation:
\[
v = 15.5 - 6(1.5)
\]
Calculate:
\[
v = 15.5 - 9 = 6.5
\]
So, \( v = 6.5 \).

Finally, we find the rental costs:
- The rental cost for each movie \( m = 1.5 \) (or $1.50).
- The rental cost for each video game \( v = 6.5 \) (or $6.50).

Thus, the rental cost for each movie is **$1.50** and for each video game is **$6.50**.