Question

question
solve for x and simplify the answer fully.
\\(\frac{x - 7}{3x + 7} = \frac{2}{9}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{x - 7}{3x+7}=\frac{2}{9}\), 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 \((x - 7)\times9=(3x + 7)\times2\).

Step2: Expand both sides

Expand the left - hand side: \(9x-63\) (using the distributive property \(a(b - c)=ab - ac\) where \(a = 9\), \(b=x\), \(c = 7\)).
Expand the right - hand side: \(6x + 14\) (using the distributive property \(a(b + c)=ab+ac\) where \(a = 2\), \(b = 3x\), \(c = 7\)).
So the equation becomes \(9x-63 = 6x+14\).

Step3: Isolate the variable terms

Subtract \(6x\) from both sides of the equation: \(9x-6x-63=6x - 6x+14\), which simplifies to \(3x-63 = 14\).

Step4: Isolate the constant term

Add 63 to both sides of the equation: \(3x-63 + 63=14 + 63\), which simplifies to \(3x=77\).

Step5: Solve for x

Divide both sides of the equation by 3: \(x=\frac{77}{3}\).

Answer:

\(x = \frac{77}{3}\)