Question

what is the solution to the equation \\(\sqrt{5x - 7} = \sqrt{3x + 5}\\)?
\\(x = 1\\)
\\(x = 12\\)
\\(x = 6\\)
\\(x = 24\\)

Explanation:

Step1: Square both sides

To eliminate the square roots, square both sides of the equation \(\sqrt{5x - 7}=\sqrt{3x + 5}\). This gives \((\sqrt{5x - 7})^2=(\sqrt{3x + 5})^2\), which simplifies to \(5x - 7 = 3x + 5\).

Step2: Solve for x

Subtract \(3x\) from both sides: \(5x-3x - 7=3x - 3x+ 5\), so \(2x - 7 = 5\). Then add 7 to both sides: \(2x-7 + 7=5 + 7\), which gives \(2x=12\). Divide both sides by 2: \(\frac{2x}{2}=\frac{12}{2}\), so \(x = 6\).

Step3: Check the solution

Substitute \(x = 6\) into the original equation. For the left - hand side: \(\sqrt{5(6)-7}=\sqrt{30 - 7}=\sqrt{23}\). For the right - hand side: \(\sqrt{3(6)+5}=\sqrt{18 + 5}=\sqrt{23}\). Since both sides are equal, \(x = 6\) is a valid solution.

Answer:

\(x = 6\)