Question

1. \\(\frac{5}{5x - 9} = \frac{4}{2x - 8}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{5}{5x - 9}=\frac{4}{2x - 8}\), we use the cross - multiplication property of proportions. If \(\frac{a}{b}=\frac{c}{d}\) (where \(b
eq0\) and \(d
eq0\)), then \(a\times d=c\times b\).
So, we get \(5\times(2x - 8)=4\times(5x - 9)\).

Step2: Expand both sides

Expand the left - hand side: \(5\times(2x-8)=5\times2x-5\times8 = 10x-40\).
Expand the right - hand side: \(4\times(5x - 9)=4\times5x-4\times9=20x - 36\).
Now our equation becomes \(10x-40 = 20x-36\).

Step3: Move the \(x\) terms to one side and constants to the other

Subtract \(10x\) from both sides: \(10x-40-10x=20x - 36-10x\), which simplifies to \(-40 = 10x-36\).
Then add 36 to both sides: \(-40 + 36=10x-36 + 36\), so \(-4=10x\).

Step4: Solve for \(x\)

Divide both sides by 10: \(x=\frac{-4}{10}=-\frac{2}{5}\).
We need to check if this solution makes the denominators non - zero.
For the denominator \(5x-9\): \(5\times(-\frac{2}{5})-9=-2 - 9=-11
eq0\).
For the denominator \(2x - 8\): \(2\times(-\frac{2}{5})-8=-\frac{4}{5}-8=-\frac{4 + 40}{5}=-\frac{44}{5}
eq0\).

Answer:

\(x =-\frac{2}{5}\)