The area of rectangle gets reduced by `8m^(2)`, when its length is reduced by 5 m and its breadth is increased by 3m. If we increase the lenght by 3m and breadth by 2m then area increase by `74 m^2` of the rectangle. Find the length and breadth of rectangle

Explanation

To solve the problem, we need to set up equations based on the information given about the rectangle's area changes. Let's denote the length of the rectangle as \( x \) meters and the breadth as \( y \) meters.

### Step 1: Set up the equations based on the area changes.

1. **First Condition**: When the length is reduced by 5 m and the breadth is increased by 3 m, the area decreases by \( 8 m^2 \).
- Original area: \( A = x \cdot y \)
- New area after changes: \( (x - 5)(y + 3) \)
- According to the condition:
\[ xy - (x - 5)(y + 3) = 8 \]
- Expanding the equation:
\[ xy - (xy + 3x - 5y - 15) = 8 \]
- Simplifying:
\[ -3x + 5y + 15 = 8 \]
- Rearranging gives us:
\[ 3x - 5y = 7 \quad \text{(Equation 1)} \]

2. **Second Condition**: When the length is increased by 3 m and the breadth is increased by 2 m, the area increases by \( 74 m^2 \).
- New area after changes: \( (x + 3)(y + 2) \)
- According to the condition:
\[ (x + 3)(y + 2) - xy = 74 \]
- Expanding the equation:
\[ xy + 2x + 3y + 6 - xy = 74 \]
- Simplifying:
\[ 2x + 3y + 6 = 74 \]
- Rearranging gives us:
\[ 2x + 3y = 68 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.

Now we have a system of linear equations:
1. \( 3x - 5y = 7 \) (Equation 1)
2. \( 2x + 3y = 68 \) (Equation 2)

We can solve these equations using substitution or elimination. Here, we will use the elimination method.

**Multiply Equation 2 by 5** to align the coefficients of \( y \):
\[ 10x + 15y = 340 \quad \text{(Equation 3)} \]

**Multiply Equation 1 by 3** to align the coefficients of \( y \):
\[ 9x - 15y = 21 \quad \text{(Equation 4)} \]

Now, add Equations 3 and 4:
\[ (10x + 15y) + (9x - 15y) = 340 + 21 \]
This simplifies to:
\[ 19x = 361 \]
Thus,
\[ x = \frac{361}{19} = 19 \]

### Step 3: Substitute \( x \) back to find \( y \).

Now substitute \( x = 19 \) back into Equation 2:
\[ 2(19) + 3y = 68 \]
This simplifies to:
\[ 38 + 3y = 68 \]
Subtracting 38 from both sides gives:
\[ 3y = 30 \]
Thus,
\[ y = \frac{30}{3} = 10 \]

### Conclusion

The length \( x \) of the rectangle is \( 19 \) meters and the breadth \( y \) is \( 10 \) meters.

### Final Answer:
- Length of the rectangle: \( 19 \) meters
- Breadth of the rectangle: \( 10 \) meters