The area of a rectangle gets reduced by 9 m$^2$ if its length is reduced by 5 m and breadth is increased by 3 m. If we increase the length by 3 m and breadth by 2 m, the area is increased by 67 m$^2$. The length of the rectangle is

### Concept & Simultaneous Linear Equations By translating the conditional changes in dimensions and area into algebraic equations, we get a system of two linear equations with two variables (length $l$ and breadth $b$). ### Step-by-Step Solution * **Condition 1:** $(l - 5)(b + 3) = lb - 9$. * Expand: $lb + 3l - 5b - 15 = lb - 9$. * Cancel $lb$ and rearrange: $3l - 5b = 6$ (Equation 1). * **Condition 2:** $(l + 3)(b + 2) = lb + 67$. * Expand: $lb + 2l + 3b + 6 = lb + 67$. * Cancel $lb$ and rearrange: $2l + 3b = 61$ (Equation 2). * To solve for $l$, we can eliminate $b$. Multiply Equation 1 by 3 and Equation 2 by 5. * $(3l - 5b = 6) \times 3 \implies 9l - 15b = 18$. * $(2l + 3b = 61) \times 5 \implies 10l + 15b = 305$. * Add the two new equations: $(9l - 15b) + (10l + 15b) = 18 + 305$. * $19l = 323$. * $l = 17$ m. ### Exam Strategy & Shortcut Substitute the given options into the first simplified condition ($3l - 5b = 6$) to see if it yields a positive, sensible value for $b$, and then verify with the second condition. If $l = 17$ (Option c): $3(17) - 5b = 6 \implies 51 - 6 = 5b \implies 5b = 45 \implies b = 9$. Check Condition 2: $2(17) + 3(9) = 34 + 27 = 61$. This matches perfectly. ### Common Pitfall Forgetting to carry over the negative signs correctly when expanding expressions like $(l - 5)(b + 3)$ is a very common source of error. ### Final Answer Therefore, the correct answer is **17 m**.