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.
[tex] \frac{-3}{4x-7} = \frac{x}{4x-3} [/tex]
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.
[tex]-3(4x-3)=x(4x-7)[/tex]
Simplify.
[tex]-12x+9=4x^2-7x[/tex]
[tex]4x^2+5x-9=0[/tex]
Factor.
[tex]4x^2-4x+9x-9=0 \\ 4x(x-1)+9(x-1)=0 \\ (4x+9)(x-1)=0[/tex]
Set each factor equal to zero and solve for x.
[tex]4x+9=0 \\ x=-9/4[/tex]
[tex]x-1=0 \\ x=1[/tex]
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 [tex]4x-7[/tex] ≠ [tex]0[/tex], so [tex]x[/tex] ≠ [tex]7/4[/tex]. For the left side, we see that [tex]4x-3[/tex] ≠ [tex]0[/tex], so [tex]x[/tex] ≠ [tex]3/4[/tex]. This means our solutions are valid.
x = -9/4, x = 1
