Chasity went on a 78 mile trip to a soccer game. On the way back, due to road construction she had to drive 13 miles per hour slower. This made the trip take 1 hour longer. How fast did she drive to the soccer game?

She drove  mph on the way to the soccer game.

Let's solve the problem using algebra.

Let \( x \) be the speed (in miles per hour) Chasity drove on the way to the soccer game.

The time taken for the trip to the game is given by:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{78}{x}
\]

On the way back, she drove 13 miles per hour slower, so her speed was \( x - 13 \) mph.

The time taken for the trip back is:

\[
\text{Time} = \frac{78}{x - 13}
\]

According to the problem, the trip back took 1 hour longer than the trip to the game. Therefore, we can write the equation:

\[
\frac{78}{x - 13} = \frac{78}{x} + 1
\]

Now, let's solve this equation for \( x \).

### Step 1: Subtract \(\frac{78}{x}\) from both sides
\[
\frac{78}{x - 13} - \frac{78}{x} = 1
\]

### Step 2: Find a common denominator and simplify
\[
\frac{78(x) - 78(x - 13)}{x(x - 13)} = 1
\]

Simplify the numerator:

\[
\frac{78x - 78x + 1014}{x(x - 13)} = 1
\]

This simplifies to:

\[
\frac{1014}{x(x - 13)} = 1
\]

### Step 3: Cross multiply and solve for \( x \)
\[
1014 = x(x - 13)
\]

Expand the right side:

\[
1014 = x^2 - 13x
\]

### Step 4: Rearrange into a standard quadratic equation
\[
x^2 - 13x - 1014 = 0
\]

### Step 5: Solve the quadratic equation using the quadratic formula
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For our equation \( x^2 - 13x - 1014 = 0 \):

- \( a = 1 \)
- \( b = -13 \)
- \( c = -1014 \)

Substitute these values into the formula:

\[
x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(1)(-1014)}}{2(1)}
\]

Simplify:

\[
x = \frac{13 \pm \sqrt{169 + 4056}}{2}
\]
\[
x = \frac{13 \pm \sqrt{4225}}{2}
\]
\[
x = \frac{13 \pm 65}{2}
\]

This gives us two possible solutions:

\[
x = \frac{13 + 65}{2} = \frac{78}{2} = 39
\]

\[
x = \frac{13 - 65}{2} = \frac{-52}{2} = -26
\]

Since speed cannot be negative, we discard \( x = -26 \).

So, \( x = 39 \).

### Final Answer:
Chasity drove **39 mph** on the way to the soccer game.

Would you like any further details or explanations?

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**Relative Questions:**
1. How long did the trip to the soccer game take?
2. What was her speed on the return trip?
3. How much longer would the return trip take if she had to drive 20 miles per hour slower instead?
4. What would her speed be if the trip took 1.5 hours longer due to road construction?
5. If the total trip time (to and from the game) was 5 hours, what were her average speeds?

**Tip:** When solving problems involving distance, speed, and time, using the formula \(\text{Distance} = \text{Speed} \times \text{Time}\) helps set up the necessary equations correctly.

Learn how to solve Chasity's speed problem involving a soccer game trip. This problem covers algebraic equations and speed, distance, time concepts. Suitable for Grades 7-9.

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