j. \( \frac{7}{x+3}=\frac{8}{x-2} \)

To solve the equation \( \frac{7}{x+3}=\frac{8}{x-2} \), you can start by cross-multiplying to eliminate the fractions. This gives you: \[ 7(x - 2) = 8(x + 3) \] Expanding both sides will yield: \[ 7x - 14 = 8x + 24 \] Now, rearranging the terms to isolate \( x \) leads to: \[ -14 - 24 = 8x - 7x \] \[ -38 = x \] So, \( x = -38 \). Now, let’s check if this solution is valid by substituting back into the original equation. If both sides balance, you've found a correct solution! --- In ancient civilizations, solving equations like this was crucial for trade and taxation, often leading to the development of algebra. The Babylonians used geometric methods to solve similar problems, long before algebra as we know it today existed! In the real world, proportions and ratios are everywhere! From cooking recipes adjusting ingredient quantities to calculating scale in blueprints or managing budget allocations, understanding how to manipulate equations is a skill that proves invaluable across multiple fields!