To solve the equation \(\frac{x}{r+a}=\frac{a}{r}\) for \(r\), we will follow a systematic approach. Let's break it down step-by-step. ### Step 1: Cross-Multiply The first step in solving the equation is to eliminate the fractions by cross-multiplying. This means we will multiply both sides of the equation by the denominators: \[ x \cdot r = a \cdot (r + a) \] ### Step 2: Distribute on the Right Side Next, we will distribute \(a\) on the right side of the equation: \[ xr = ar + a^2 \] ### Step 3: Rearrange the Equation Now, we want to isolate \(r\). To do this, we will move all terms involving \(r\) to one side and the constant terms to the other side. We can subtract \(ar\) from both sides: \[ xr - ar = a^2 \] ### Step 4: Factor Out \(r\) Now, we can factor \(r\) out of the left side: \[ r(x - a) = a^2 \] ### Step 5: Solve for \(r\) Finally, we will divide both sides by \((x - a)\) to solve for \(r\): \[ r = \frac{a^2}{x - a} \] ### Conclusion Thus, the correct expression for \(r\) is: \[ r = \frac{a^2}{x - a} \] ### Answer Verification Now, let's check the options provided: - **A. \(\frac{a^{2}}{(x-a)}\)** - This matches our derived formula. - **B. \(\frac{a^{2}}{(x+a)}\)** - This is incorrect because it has \(x + a\) instead of \(x - a\). - **C. \(\frac{a}{x-a}\)** - This is incorrect because it does not have \(a^2\) in the numerator. - **D. \(\frac{a}{x+a}\)** - This is incorrect for the same reasons as option B and also does not have \(a^2\). ### Summary of Key Points - We started with the equation \(\frac{x}{r+a}=\frac{a}{r}\) and cross-multiplied to eliminate fractions. - We rearranged the equation to isolate terms involving \(r\) and factored \(r\) out. - The final expression for \(r\) is \(\frac{a^2}{x - a}\), which corresponds to option A. - Always check each option against your derived formula to confirm correctness. ### Revision Summary - Cross-multiply to eliminate fractions. - Rearrange the equation to isolate the variable. - Factor out the variable to simplify. - Verify your answer against the provided options to ensure accuracy.