\( \frac{15}{a^{2}-1}=\frac{5}{2 a-2} \) Which equation results from cross-multiplying?

To find the equation that results from cross-multiplying the given equation \( \frac{15}{a^{2}-1}=\frac{5}{2a-2} \), we need to cross-multiply the fractions. Cross-multiplying means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. So, cross-multiplying the given equation gives us: \[ 15(2a-2) = 5(a^{2}-1) \] Now, we can simplify this equation to find the resulting equation. Solve the equation \( 15(2a-2) = 5(a^{2}-1) \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(15\left(2a-2\right)=5\left(a^{2}-1\right)\) - step1: Swap the sides: \(5\left(a^{2}-1\right)=15\left(2a-2\right)\) - step2: Expand the expression: \(5a^{2}-5=30a-30\) - step3: Move the expression to the left side: \(5a^{2}+25-30a=0\) - step4: Factor the expression: \(5\left(a-5\right)\left(a-1\right)=0\) - step5: Divide the terms: \(\left(a-5\right)\left(a-1\right)=0\) - step6: Separate into possible cases: \(\begin{align}&a-5=0\\&a-1=0\end{align}\) - step7: Solve the equation: \(\begin{align}&a=5\\&a=1\end{align}\) - step8: Rewrite: \(a_{1}=1,a_{2}=5\) The equation that results from cross-multiplying the given equation is \( a = 1 \) or \( a = 5 \).