Question

solve for h.
\\(\frac{h + 7}{h - 2} = \frac{h - 9}{h + 6}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{h + 7}{h - 2}=\frac{h - 9}{h + 6}\), we use the cross - multiplication property of proportions. If \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\). So we have \((h + 7)(h + 6)=(h - 9)(h - 2)\).

Step2: Expand both sides

Expand the left - hand side using the FOIL method: \((h + 7)(h + 6)=h\times h+h\times6 + 7\times h+7\times6=h^{2}+6h + 7h+42=h^{2}+13h + 42\).

Expand the right - hand side using the FOIL method: \((h - 9)(h - 2)=h\times h+h\times(-2)+(-9)\times h+(-9)\times(-2)=h^{2}-2h-9h + 18=h^{2}-11h + 18\).

Step3: Subtract \(h^{2}\) from both sides

Subtract \(h^{2}\) from both sides of the equation \(h^{2}+13h + 42=h^{2}-11h + 18\). We get \(h^{2}-h^{2}+13h + 42=h^{2}-h^{2}-11h + 18\), which simplifies to \(13h + 42=-11h + 18\).

Step4: Add \(11h\) to both sides

Add \(11h\) to both sides of the equation \(13h + 42=-11h + 18\). We have \(13h+11h + 42=-11h + 11h+18\), so \(24h + 42=18\).

Step5: Subtract 42 from both sides

Subtract 42 from both sides of the equation \(24h + 42=18\). We get \(24h+42 - 42=18 - 42\), which simplifies to \(24h=-24\).

Step6: Divide both sides by 24

Divide both sides of the equation \(24h=-24\) by 24. \(\frac{24h}{24}=\frac{-24}{24}\), so \(h=-1\).

We need to check if this solution makes the original denominators zero. For the original equation, the denominators are \(h - 2\) and \(h + 6\). When \(h=-1\), \(h - 2=-1-2=-3
eq0\) and \(h + 6=-1 + 6=5
eq0\). So \(h=-1\) is a valid solution.

Answer:

\(h=-1\)