Question

Step-by-step solution to:D%7B4x%20-%203%7D%20" id="TexFormula1" title=" - \frac{3}{4x - 7} = \frac{x}{4x - 3} " alt=" - \frac{3}{4x - 7} = \frac{x}{4x - 3} " align="absmiddle" class="latex-formula">

Answer 1

The key to solving this problem is recognizing it is a proportion and can be solved using the technique called cross-multiplication. This means that you are multiplying the numerator of each side by the denominator of the other side.

$\frac{-3}{4x-7} = \frac{x}{4x-3}$

You multiply the numerator of the first side by the denominator of the second and the denominator of the first by the numerator of the second.

$-3(4x-3)=x(4x-7)$

Simplify.

$-12x+9=4x^2-7x$

$4x^2+5x-9=0$

Factor.
$4x^2-4x+9x-9=0 \\ 4x(x-1)+9(x-1)=0 \\ (4x+9)(x-1)=0$

Set each factor equal to zero and solve for x.

$4x+9=0 \\ x=-9/4$

$x-1=0 \\ x=1$

Finally, you want to look at the original expression and make sure that these solutions don't violate any restrictions on division by zero.

For the right side, we see that $4x-7$ ≠ $0$, so $x$ ≠ $7/4$. For the left side, we see that $4x-3$ ≠ $0$, so $x$ ≠ $3/4$. This means our solutions are valid.

x = -9/4, x = 1