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

Step 1: Define variables and form the first equation

We start by defining variables for the distance, speed, and time. The fundamental relationship is distance equals speed multiplied by time. The first condition states that if the train were $10\text{ km/h}$ faster, it would take $2\text{ hours}$ less. We use this to form our second equation.

Step 2: Form the second equation

The second condition states that if the train were $10\text{ km/h}$ slower, it would have taken $3\text{ hours}$ more than the scheduled time. We use this information to form our third equation.

Step 3: Simplify the equations

Now we substitute $d = vt$ into equations (2) and (3) and simplify them. This will give us two linear equations in terms of $v$ and $t$.

Step 4: Solve the system of linear equations

We now have a system of two linear equations with two variables, $v$ and $t$. We can solve this system using the elimination method. Multiplying equation (4) by 2 makes the coefficient of $t$ the same in both equations, allowing us to eliminate $t$ by subtraction. Once $v$ is found, we substitute it back into one of the simplified equations to find $t$.

Step 5: Calculate the distance

Finally, with the values of speed ($v$) and time ($t$) determined, we can calculate the total distance covered by the train using the original formula $d = v \times t$.

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