To solve the equation [tex]\(\frac{-7}{x} = \frac{-6}{3 x} + 5\)[/tex], follow these steps:
1. Simplify the right-hand side of the equation:
Start with the equation:
[tex]\[
\frac{-7}{x} = \frac{-6}{3 x} + 5
\][/tex]
The term [tex]\(\frac{-6}{3 x}\)[/tex] can be simplified. Notice that [tex]\(\frac{-6}{3 x}\)[/tex] is the same as [tex]\(\frac{-6}{3} \cdot \frac{1}{x}\)[/tex]:
[tex]\[
\frac{-6}{3 x} = \frac{-6}{3} \cdot \frac{1}{x} = -2 \cdot \frac{1}{x} = \frac{-2}{x}
\][/tex]
Therefore, the original equation becomes:
[tex]\[
\frac{-7}{x} = \frac{-2}{x} + 5
\][/tex]
2. Eliminate the denominator:
To eliminate the fraction, multiply both sides of the equation by [tex]\(x\)[/tex]:
[tex]\[
x \cdot \frac{-7}{x} = x \cdot \left( \frac{-2}{x} + 5 \right)
\][/tex]
Simplifying this, we get:
[tex]\[
-7 = -2 + 5x
\][/tex]
3. Isolate [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], first move all the constant terms to one side of the equation:
[tex]\[
-7 + 2 = 5x
\][/tex]
Simplify the left-hand side:
[tex]\[
-5 = 5x
\][/tex]
Divide both sides by 5:
[tex]\[
x = \frac{-5}{5}
\][/tex]
Thus:
[tex]\[
x = -1
\][/tex]
So, the solution to the equation [tex]\(\frac{-7}{x} = \frac{-6}{3 x} + 5\)[/tex] is:
[tex]\[
x = -1
\][/tex]
