Question

. solve for x and simplify the answer fully.
\\(\frac{4}{5x + 6} = \frac{9}{4x + 5}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{4}{5x + 6}=\frac{9}{4x+5}\), we use the cross - multiplication property of proportions. If \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\). So we have \(4\times(4x + 5)=9\times(5x + 6)\).

Step2: Expand both sides

Expand the left - hand side: \(4\times(4x+5)=16x + 20\).
Expand the right - hand side: \(9\times(5x + 6)=45x+54\).
Now our equation becomes \(16x + 20 = 45x+54\).

Step3: Move the x - terms to one side and constants to the other

Subtract \(16x\) from both sides: \(16x-16x + 20=45x-16x + 54\), which simplifies to \(20 = 29x+54\).
Then subtract 54 from both sides: \(20 - 54=29x+54 - 54\), so \(- 34 = 29x\).

Step4: Solve for x

Divide both sides by 29: \(x=\frac{-34}{29}\)

Answer:

\(x =-\frac{34}{29}\)