Question

A cyclist cycles for x hours at 12km/h and y hours at 18km/h. Altogether she cycles 78km in 5 hours. Find x and y

Answer 1

To solve for the number of hours \( x \) and \( y \) that the cyclist spends cycling at 12 km/h and 18 km/h, respectively, we can set up a system of equations based on the information provided.

1. We know that the total time spent cycling is 5 hours:
\[
x + y = 5
\]

2. We also know that the total distance cycled is 78 km. The distance cycled at each speed can be calculated as:
- Distance at 12 km/h: \( 12x \)
- Distance at 18 km/h: \( 18y \)

Therefore, we can write the second equation for the total distance:
\[
12x + 18y = 78
\]

Now we have the following system of equations:

1. \( x + y = 5 \)
2. \( 12x + 18y = 78 \)

We can solve this system step by step.

**Step 1:** Solve the first equation for \( y \):
\[
y = 5 - x
\]

**Step 2:** Substitute \( y \) into the second equation:
\[
12x + 18(5 - x) = 78
\]

**Step 3:** Distribute the 18:
\[
12x + 90 - 18x = 78
\]

**Step 4:** Combine like terms:
\[
-6x + 90 = 78
\]

**Step 5:** Rearrange to solve for \( x \):
\[
-6x = 78 - 90
\]
\[
-6x = -12
\]
\[
x = \frac{-12}{-6} = 2
\]

**Step 6:** Substitute \( x \) back to find \( y \):
\[
y = 5 - x = 5 - 2 = 3
\]

Thus, the solution is:
\[
\boxed{x = 2} \quad \text{and} \quad \boxed{y = 3}
\]

### Verification:
- Time spent cycling at 12 km/h: \( x = 2 \) hours, so distance \( = 12 \times 2 = 24 \) km.
- Time spent cycling at 18 km/h: \( y = 3 \) hours, so distance \( = 18 \times 3 = 54 \) km.
- Total distance: \( 24 + 54 = 78 \) km.
- Total time: \( 2 + 3 = 5 \) hours.

Both conditions are satisfied, confirming that the solution is correct.