To solve the equation
[tex]\[
\frac{6}{x+5} = \frac{3}{x-10},
\][/tex]
we’ll follow a systematic approach:
1. Cross-Multiply:
By cross-multiplying the given equation, we eliminate the fractions. Doing so, we get
[tex]\[
6(x - 10) = 3(x + 5).
\][/tex]
2. Expand Both Sides:
Next, distribute the constants on each side:
[tex]\[
6x - 60 = 3x + 15.
\][/tex]
3. Isolate the Variable:
To isolate [tex]\(x\)[/tex], we need to combine like terms. First, subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
6x - 3x - 60 = 15,
\][/tex]
which simplifies to:
[tex]\[
3x - 60 = 15.
\][/tex]
Next, add 60 to both sides:
[tex]\[
3x = 75.
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Finally, divide both sides by 3:
[tex]\[
x = 25.
\][/tex]
So, the solution to the equation
[tex]\[
\frac{6}{x+5} = \frac{3}{x-10}
\][/tex]
is:
[tex]\[
x = 25.
\][/tex]
After verifying the potential solutions, only [tex]\(x = 25\)[/tex] satisfies the given equation. Therefore, the correct solution is [tex]\(x = 25\)[/tex] only.
