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.