How do you solve \frac{x + 7}{x} = \frac{7}{9}?

Mar 5, 2018

$x = - \frac{63}{2} \mathmr{and} - 31.5$

Explanation:

$\frac{x + 7}{x} = \frac{7}{9}$

There may be more complex, quicker ways to solve this, but by simply moving around the numbers in order to isolate $x$, we can solve this question.

$\frac{x + 7}{x} \times 9 = \frac{7}{\textcolor{red}{\cancel{\textcolor{b l a c k}{9}}}} \textcolor{red}{\cancel{\times 9}}$

$\frac{\left(x + 7\right) \times 9}{x} = 7$

$\frac{9 x + 63}{\textcolor{red}{\cancel{\textcolor{b l a c k}{x}}}} \textcolor{red}{\cancel{\times x}} = 7 \textcolor{red}{\times x}$

$9 x + 63 \textcolor{red}{- 7 x} = \textcolor{red}{\cancel{\textcolor{b l a c k}{7 x} - 7 x}}$

$2 x \textcolor{red}{\cancel{\textcolor{b l a c k}{+ 63} - 63}} = 0 \textcolor{red}{- 63}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} x}{\textcolor{red}{\cancel{2}}} = - \frac{63}{\textcolor{red}{2}}$

$\textcolor{b l u e}{x = - \frac{63}{2}}$

$\textcolor{b l u e}{x = - 31.5}$

$\underline{\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times}}$

We can check this by substituting the result for the Pronumeral.

$\frac{x + 7}{x} = \frac{7}{9}$

$\frac{- 31.5 + 7}{-} 31.5 = \frac{7}{9}$

$\frac{24.5 \textcolor{red}{\times 2}}{31.5 \textcolor{red}{\times 2}} = \frac{7}{9}$

$\frac{49 \textcolor{red}{\div 7}}{63 \textcolor{red}{\div 7}} = \frac{7}{9}$

$\frac{7}{9} = \frac{7}{9}$

$\underline{\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times}}$