Question

solve the following. check

1. $\frac{h+7}{6}=\frac{h-3}{h-6}$

Explanation:

Step1: Cross-multiply to eliminate fractions

$(h+7)(h-6) = 6(h-3)$

Step2: Expand both sides

$h^2 -6h +7h -42 = 6h -18$

Step3: Simplify left side

$h^2 + h -42 = 6h -18$

Step4: Rearrange to quadratic form

$h^2 + h -42 -6h +18 = 0$
$h^2 -5h -24 = 0$

Step5: Factor quadratic equation

$(h-8)(h+3) = 0$

Step6: Solve for h

$h-8=0 \quad \text{or} \quad h+3=0$

Step7: Check for extraneous solutions

For $h=8$: $\frac{8+7}{6}=\frac{15}{6}=2.5$, $\frac{8-3}{8-6}=\frac{5}{2}=2.5$ (valid)
For $h=-3$: $\frac{-3+7}{6}=\frac{4}{6}=\frac{2}{3}$, $\frac{-3-3}{-3-6}=\frac{-6}{-9}=\frac{2}{3}$ (valid)

Answer:

$h=8$ or $h=-3$