Question

1. \\(\frac{y}{4} = \frac{9}{y}\\)

Explanation:

Step1: Cross - multiply the equation

Given the equation \(\frac{y}{4}=\frac{9}{y}\), cross - multiplying (which is based on the property that if \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d = b\times c\)) gives us \(y\times y=4\times9\).

Step2: Simplify the equation

Simplifying the left - hand side and the right - hand side, we get \(y^{2}=36\).

Step3: Solve for y

To solve for \(y\), we take the square root of both sides. Remember that if \(x^{2}=a\) (\(a\geq0\)), then \(x = \pm\sqrt{a}\). So, \(y=\pm\sqrt{36}=\pm6\). We need to check these solutions in the original equation.

• For \(y = 6\): Left - hand side \(\frac{6}{4}=\frac{3}{2}\), Right - hand side \(\frac{9}{6}=\frac{3}{2}\). So, \(y = 6\) is a valid solution.
• For \(y=- 6\): Left - hand side \(\frac{-6}{4}=-\frac{3}{2}\), Right - hand side \(\frac{9}{-6}=-\frac{3}{2}\). So, \(y=-6\) is also a valid solution.

Answer:

\(y = 6\) or \(y=-6\)