Question

solve for x and simplify the answer fully.
\\(\frac{6}{x + 5} = \frac{7}{x - 7}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{6}{x + 5}=\frac{7}{x - 7}\), 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 \(6(x - 7)=7(x + 5)\).

Step2: Expand both sides

Expand the left - hand side: \(6(x - 7)=6x-42\) (using the distributive property \(a(b - c)=ab - ac\) where \(a = 6\), \(b=x\), and \(c = 7\)).
Expand the right - hand side: \(7(x + 5)=7x + 35\) (using the distributive property \(a(b + c)=ab+ac\) where \(a = 7\), \(b = x\), and \(c = 5\)).
So our equation becomes \(6x-42=7x + 35\).

Step3: Isolate the variable x

Subtract \(6x\) from both sides of the equation:
\(6x-42-6x=7x + 35-6x\)
Simplifying both sides, we get \(-42=x + 35\).
Then subtract 35 from both sides:
\(-42-35=x+35 - 35\)
\(-77=x\)

Answer:

\(x=-77\)