Question

solve the equation \\(\frac{2}{x - 3} = \frac{3}{x}\\)

Explanation:

Step1: Cross - multiply the equation

Given the equation \(\frac{2}{x - 3}=\frac{3}{x}\), cross - multiplying (by the property that if \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\)) gives us \(2\times x = 3\times(x - 3)\).

Step2: Expand the right - hand side

Expanding the right - hand side of the equation \(2x=3(x - 3)\), we use the distributive property \(a(b - c)=ab - ac\). So, \(2x = 3x-9\).

Step3: Solve for \(x\)

Subtract \(2x\) from both sides of the equation \(2x = 3x - 9\):
\(2x-2x=3x - 2x-9\)
\(0=x - 9\)
Then add 9 to both sides: \(x=9\).
We need to check if this solution is valid. The original equation has denominators \(x-3\) and \(x\). When \(x = 9\), \(x-3=9 - 3=6
eq0\) and \(x = 9
eq0\), so \(x = 9\) is a valid solution.

Answer:

\(x = 9\)