Question

solve for x and simplify the answer fully.
\\(\frac{x + 5}{3} = \frac{x - 5}{2}\\)

Explanation:

Step1: Cross - multiply to eliminate fractions

To solve the equation \(\frac{x + 5}{3}=\frac{x - 5}{2}\), we cross - multiply. Cross - multiplying gives us \(2(x + 5)=3(x - 5)\). This is based on the property that if \(\frac{a}{b}=\frac{c}{d}\) (where \(b
eq0\) and \(d
eq0\)), then \(a\times d=b\times c\).

Step2: Expand both sides

Expand the left - hand side: \(2(x + 5)=2x+10\) (using the distributive property \(a(b + c)=ab+ac\) with \(a = 2\), \(b=x\), and \(c = 5\)).
Expand the right - hand side: \(3(x - 5)=3x-15\) (using the distributive property \(a(b - c)=ab - ac\) with \(a = 3\), \(b=x\), and \(c = 5\)).
So our equation becomes \(2x+10 = 3x-15\).

Step3: Isolate the variable \(x\)

Subtract \(2x\) from both sides of the equation: \(2x+10-2x=3x - 15-2x\).
Simplifying both sides, we get \(10=x - 15\).
Then add 15 to both sides: \(10 + 15=x-15 + 15\).
Simplifying, we find that \(x=25\).

Answer:

\(x = 25\)