# How do you solve and check your solutions to 8/5x=4/15?

May 21, 2018

See a solution process below:

#### Explanation:

Multiply each side of the equation by $\frac{\textcolor{red}{5}}{\textcolor{b l u e}{8}}$ to solve for $x$ while keeping the equation balanced:

$\frac{\textcolor{red}{5}}{\textcolor{b l u e}{8}} \times \frac{8}{5} x = \frac{\textcolor{red}{5}}{\textcolor{b l u e}{8}} \times \frac{4}{15}$

$\frac{\cancel{\textcolor{red}{5}}}{\cancel{\textcolor{b l u e}{8}}} \times \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{8}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}}} x = \frac{\cancel{\textcolor{red}{5}} \textcolor{red}{1}}{\cancel{\textcolor{b l u e}{8}} \textcolor{b l u e}{2}} \times \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{4}}} 1}{\textcolor{red}{\cancel{\textcolor{b l a c k}{15}}} 3}$

$x = \frac{\textcolor{red}{1}}{\textcolor{b l u e}{2}} \times \frac{1}{3}$

$x = \frac{1}{6}$

To check the solution substitute $\frac{1}{6}$ for $x$ in the original equation and ensure both sides of the equation are equal:

$\frac{8}{5} x = \frac{4}{15}$ becomes:

$\frac{8}{5} \times \frac{1}{6} = \frac{4}{15}$

$\frac{8 \times 1}{5 \times 6} = \frac{4}{15}$

$\frac{8}{30} = \frac{4}{15}$

$\frac{2 \times 4}{2 \times 15} = \frac{4}{15}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \times 4}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \times 15} = \frac{4}{15}$

$\frac{4}{15} = \frac{4}{15}$

May 21, 2018

Solution: $\setminus \frac{1}{6}$

Check: see below.

#### Explanation:

Any equation of the form

$a x = b$

yields the solution

$x = \frac{b}{a}$

In your case, $a = \frac{8}{5}$ and $b = \frac{4}{15}$

So, we must divide by $a$, which is a fraction. So, $\frac{b}{a}$ is the same thing as $b \setminus \cdot {a}^{- 1}$, where ${a}^{- 1}$ is the inverse fraction of $a$, obtained by switching numerator and denominator. So, the solution is

$x = \frac{b}{a} = \setminus \frac{\frac{4}{15}}{\frac{8}{5}} = \setminus \frac{\cancel{4}}{\cancel{15} 3} \setminus \cdot \setminus \frac{\cancel{5}}{\cancel{8} 2} = \frac{1}{6}$

To check the value, simply plug it in, substituting $x$:

$\setminus \frac{8}{5} \setminus \textcolor{g r e e n}{x} = \setminus \frac{4}{15} \setminus \implies \setminus \frac{\cancel{8} 4}{5} \setminus \textcolor{g r e e n}{\setminus \frac{1}{\cancel{6} 3}} = \setminus \frac{4}{15}$

which is true, since the left hand side evaluates to $\setminus \frac{4}{15}$