Question

question
solve for all values of x.
\\(\frac{x + 7}{x - 1} = \frac{3}{x}\\)
answer attempt 1 out of 2
additional solution no solution

Explanation:

Step1: Cross-multiply to eliminate fractions

$x(x + 7) = 3(x - 1)$

Step2: Expand both sides

$x^2 + 7x = 3x - 3$

Step3: Rearrange to standard quadratic form

$x^2 + 7x - 3x + 3 = 0$
$x^2 + 4x + 3 = 0$

Step4: Factor the quadratic

$(x + 1)(x + 3) = 0$

Step5: Solve for x

Set each factor equal to 0:
$x + 1 = 0 \implies x = -1$
$x + 3 = 0 \implies x = -3$

Step6: Verify no extraneous solutions

Substitute $x=-1$: $\frac{-1+7}{-1-1}=\frac{6}{-2}=-3$, $\frac{3}{-1}=-3$ (valid)
Substitute $x=-3$: $\frac{-3+7}{-3-1}=\frac{4}{-4}=-1$, $\frac{3}{-3}=-1$ (valid)

Answer:

$x = -1$ and $x = -3$