Question

question
solve for x and simplify the answer fully.
\\(\frac{3}{5x - 3} = \frac{5}{x + 9}\\)

Explanation:

Step1: Cross - multiply to eliminate fractions

To solve the equation \(\frac{3}{5x - 3}=\frac{5}{x + 9}\), we cross - multiply. Cross - multiplying gives us \(3(x + 9)=5(5x - 3)\). The reason for cross - multiplying is that if \(\frac{a}{b}=\frac{c}{d}\) (where \(b
eq0\) and \(d
eq0\)), then \(a\times d=c\times b\).

Step2: Expand both sides of the equation

Expanding the left - hand side: \(3(x + 9)=3x+27\) (using the distributive property \(a(b + c)=ab+ac\) with \(a = 3\), \(b=x\), and \(c = 9\)).
Expanding the right - hand side: \(5(5x - 3)=25x-15\) (using the distributive property \(a(b - c)=ab - ac\) with \(a = 5\), \(b = 5x\), and \(c = 3\)).
So our equation becomes \(3x + 27=25x-15\).

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

Subtract \(3x\) from both sides: \(3x+27-3x=25x - 15-3x\), which simplifies to \(27 = 22x-15\).
Then add 15 to both sides: \(27 + 15=22x-15 + 15\), so \(42=22x\).

Step4: Solve for \(x\)

Divide both sides by 22: \(x=\frac{42}{22}\). We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So \(x=\frac{21}{11}\). We also need to check that the solution does not make the denominators of the original equation zero.
For the denominator \(5x-3\): when \(x = \frac{21}{11}\), \(5\times\frac{21}{11}-3=\frac{105}{11}-\frac{33}{11}=\frac{72}{11}
eq0\).
For the denominator \(x + 9\): when \(x=\frac{21}{11}\), \(\frac{21}{11}+9=\frac{21 + 99}{11}=\frac{120}{11}
eq0\). So the solution is valid.

Answer:

\(x=\frac{21}{11}\)