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 the problem, we can set up a system of equations based on the information provided.

1. The total time cycling is given by:
\[
x + y = 5
\]
(where \(x\) is the time spent cycling at 12 km/h and \(y\) is the time spent cycling at 18 km/h).

2. The total distance cycled is given by:
\[
12x + 18y = 78
\]
(the distance covered at 12 km/h is \(12x\) and the distance covered at 18 km/h is \(18y\)).

Now we have a system of equations:
\[
\begin{align*}
1) & \quad x + y = 5 \\
2) & \quad 12x + 18y = 78
\end{align*}
\]

We can solve this system step by step:

From the first equation, we can express \(y\) in terms of \(x\):
\[
y = 5 - x
\]

Now, substitute \(y\) into the second equation:
\[
12x + 18(5 - x) = 78
\]

Expanding this yields:
\[
12x + 90 - 18x = 78
\]

Combining like terms:
\[
-6x + 90 = 78
\]

Subtract 90 from both sides:
\[
-6x = -12
\]

Dividing by -6:
\[
x = 2
\]

Now, substitute \(x\) back into the equation for \(y\):
\[
y = 5 - x = 5 - 2 = 3
\]

Therefore, the cyclist cycled for:
\[
\boxed{x = 2} \text{ hours at 12 km/h and } \boxed{y = 3} \text{ hours at 18 km/h.}
\]

To verify:
- Total time: \(2 + 3 = 5\) hours (correct)
- Total distance: \(12(2) + 18(3) = 24 + 54 = 78\) km (correct).

Everything checks out, so the final answer is \(x = 2\) hours and \(y = 3\) hours.