Question

question
solve for x and simplify the answer fully.
\\(\frac{8}{x - 6} = \frac{5}{x + 6}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{8}{x - 6}=\frac{5}{x + 6}\), we cross - multiply. Cross - multiplying gives us \(8(x + 6)=5(x - 6)\). The reason for cross - multiplying is that if \(\frac{a}{b}=\frac{c}{d}\), then \(ad = bc\) (where \(b
eq0\) and \(d
eq0\)).

Step2: Expand both sides of the equation

Expanding the left - hand side: \(8(x + 6)=8x+48\) (using the distributive property \(a(b + c)=ab+ac\) with \(a = 8\), \(b=x\), and \(c = 6\)).
Expanding the right - hand side: \(5(x - 6)=5x-30\) (using the distributive property \(a(b - c)=ab - ac\) with \(a = 5\), \(b=x\), and \(c = 6\)).
So our equation becomes \(8x + 48=5x-30\).

Step3: Isolate the variable x

Subtract \(5x\) from both sides of the equation: \(8x-5x + 48=5x-5x-30\), which simplifies to \(3x + 48=-30\).
Then subtract 48 from both sides: \(3x+48 - 48=-30 - 48\), which gives \(3x=-78\).

Step4: Solve for x

Divide both sides of the equation \(3x=-78\) by 3: \(\frac{3x}{3}=\frac{-78}{3}\), so \(x=-26\).
We need to check if this solution makes the original denominators zero. For \(x=-26\), \(x - 6=-26-6=-32
eq0\) and \(x + 6=-26 + 6=-20
eq0\). So \(x=-26\) is a valid solution.

Answer:

\(x=-26\)