# Ex.3.7 Q3 Pair of Linear Equations in Two Variables Solution - NCERT Maths Class 10

## Question

A train covered a certain distance at a uniform speed. If the train would have been \(10\,\rm{ km/h}\) faster, it would have taken \(2\) hours less than the scheduled time. And if the train were slower by \(10\,\rm{ km/h;}\) it would have taken \(3\) hours more than the scheduled time. Find the distance covered by the train.

## Text Solution

**What is Known?**

Changes in speed of the train as well in the time.

**What is Unknown?**

Distance covered by the train.

**Reasoning:**

Assuming uniform speed of the train be \(x\,{\rm{km/h }}\) and time taken to travel a given distance be \(t\) hours. Then distance can be calculated by;

\[{\text{Distance}} = {\rm{ Speed }} \times {\rm{ Time}}\]

**Steps:**

Let the uniform speed of the train be \(x\,{\rm{ km/h}}\) and the scheduled time to travel the given distance be \(t\) hours

Then the distance be \(xt\,{\rm{ km}}\)

When the train would have been \(10\,{\rm{ km/h}}\) faster, it would have taken \(2\) hours less than the scheduled time;

\[\begin{align}\left( {x + 10} \right)\left( {t - 2} \right)& = xt\\xt - 2x + 10t - 20 &= xt\\ - 2x + 10t &= 20 \qquad \quad \left( 1 \right)\end{align}\]

When the train were slower by \(10\,{\rm{ km/h,}}\) it would have taken \(2\) hours\] more than the scheduled time;

\[\begin{align}\left( {x - 10} \right)\left( {t + 3} \right) &= xt\\xt + 3x - 10t - 30 &= xt\\3x - 10t &= 30 \qquad \qquad \left( 2 \right)\end{align}\]

Adding equations \((1)\) and \((2),\) we obtain

\[x = 50\]

Substituting \(x = 50\) in equation \((1),\) we obtain

\[\begin{align} - 2 \times 50 + 10t &= 20\\- 100 + 10t &= 20\\10t &= 120\\t &= \frac{{120}}{{10}}\\t &= 12

\end{align}\]

Therefore, distance, \(xt = 50 \times 12 = 600\)

Hence, the distance covered by the train is \(600 \,\rm{km.}\)