Question

LaTeX: 1-\frac{7}{x}=\frac{-6}{x^2}

Answer 1

To solve the equation

\[
1 - \frac{7}{x} = \frac{-6}{x^2},
\]

first, we can eliminate the fractions by multiplying both sides by \(x^2\) (assuming \(x \neq 0\)):

\[
x^2 \left(1 - \frac{7}{x}\right) = -6.
\]

Expanding the left side gives:

\[
x^2 - 7x = -6.
\]

Next, we rearrange the equation to set it to zero:

\[
x^2 - 7x + 6 = 0.
\]

Now we can factor this quadratic equation:

\[
(x - 6)(x - 1) = 0.
\]

Setting each factor equal to zero gives us the solutions:

\[
x - 6 = 0 \quad \Rightarrow \quad x = 6,
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1.
\]

Thus, the solutions to the equation \(1 - \frac{7}{x} = \frac{-6}{x^2}\) are

\[
x = 1 \quad \text{and} \quad x = 6.
\]