Question

What is the solution of the equation \(\frac{9}{x+5}=\frac{7}{x-5} ?\) (A) 5 (B) 8 (C) 20 (D) 40 (E) 80

Short answer

The solution for x is 40, hence answer choice (D) is correct.

Step-by-step solution

Identify the extremes and means

First, identify the extremes (a and d) and means (b and c) in the proportion. From the given equation, \( \frac{9}{x+5}=\frac{7}{x-5}\), the extremes are 9 and \( x - 5 \) while the means are \( x+5 \) and 7.

Apply cross-multiplication

In a proportion a/b=c/d, the product of the means equals the product of the extremes. That is, a x d = c x b. Apply this here: 9 x (x - 5) = 7 x (x + 5). This will solve to 9x - 45 = 7x + 35.

Simplify the equation

Simplify the equation by substracting 7x from both sides to get 2x - 45 = 35. Then, add 45 to both sides to get 2x = 80.

Solve for x

Finally, divide both sides by 2 to solve for x. The solution is x = 40.