A train covered a certain distance at a uniform speed. If the train had been 5 kmph faster, it would have taken 3 hours less than the scheduled time. And, if the train were slower by 4 kmph, it would have taken 3 hours more than the scheduled time. Find the length of the journey.

Explanation

To solve the problem step by step, we will define the variables and set up the equations based on the information given in the question.

### Step 1: Define Variables
Let:
- \( x \) = speed of the train in km/h
- \( y \) = time taken for the journey in hours

The distance covered by the train can be expressed as:
\[ \text{Distance} = \text{Speed} \times \text{Time} = x \cdot y \]

### Step 2: Set Up the First Equation
According to the problem, if the train had been 5 km/h faster, it would have taken 3 hours less. Therefore, the new speed is \( x + 5 \) and the new time is \( y - 3 \). The distance remains the same, so we can set up the equation:
\[ x \cdot y = (x + 5)(y - 3) \]

Expanding this equation:
\[ x \cdot y = xy - 3x + 5y - 15 \]
Rearranging gives us:
\[ 0 = -3x + 5y - 15 \]
Thus, we can write the first equation as:
\[ 3x - 5y + 15 = 0 \] (Equation 1)

### Step 3: Set Up the Second Equation
Now, if the train were slower by 4 km/h, it would have taken 3 hours more. The new speed is \( x - 4 \) and the new time is \( y + 3 \). Again, the distance remains the same:
\[ x \cdot y = (x - 4)(y + 3) \]

Expanding this equation:
\[ x \cdot y = xy + 3x - 4y - 12 \]
Rearranging gives us:
\[ 0 = 3x - 4y - 12 \]
Thus, we can write the second equation as:
\[ 3x - 4y - 12 = 0 \] (Equation 2)

### Step 4: Solve the Equations
Now we have two equations:
1. \( 3x - 5y + 15 = 0 \)
2. \( 3x - 4y - 12 = 0 \)

We can subtract Equation 2 from Equation 1:
\[ (3x - 5y + 15) - (3x - 4y - 12) = 0 \]
This simplifies to:
\[ -5y + 4y + 15 + 12 = 0 \]
\[ -y + 27 = 0 \]
Thus, we find:
\[ y = 27 \text{ hours} \]

### Step 5: Substitute Back to Find Speed
Now we can substitute \( y = 27 \) back into one of the original equations to find \( x \). Using Equation 1:
\[ 3x - 5(27) + 15 = 0 \]
\[ 3x - 135 + 15 = 0 \]
\[ 3x - 120 = 0 \]
\[ 3x = 120 \]
\[ x = 40 \text{ km/h} \]

### Step 6: Find the Length of the Journey
Now that we have both \( x \) and \( y \), we can find the length of the journey:
\[ \text{Distance} = x \cdot y = 40 \cdot 27 = 1080 \text{ km} \]

### Final Answer
The length of the journey is **1080 km**.