# Solving One-Step Linear Equations with Mixed Numbers: Multiplication

A
**
mixed number
**
is a number expressed as the sum of a
whole number
and a
fractions
, such as
$3\frac{1}{4}$
.

It is usually easier to do calculations with improper fractions than mixed numbers, but mixed numbers give a better idea of the size of a number. So you should know how to convert back and forth.

To solve an equation that has a mixed number coefficient, we convert the mixed number to an improper fraction as the first step.

We can solve linear equations when only multiplication or division is involved. If there's a coefficient in front of the variable, multiply by the reciprocal of that number to get a coefficient of $1$ .

**
Example :
**

Solve.
*
*

$1\frac{1}{4}x=1\frac{1}{9}$

Rewrite the mixed numbers as improper fractions.

$\frac{5}{4}x=\frac{10}{9}$

The inverse operation of division is multiplication.

To isolate the variable $x$ (to get a coefficient of $1$ ), multiply both sides by $\frac{4}{5}$ which is the reciprocal of $\frac{5}{4}$ .

$\frac{4}{5}\cdot \left(\frac{5}{4}x\right)=\frac{10}{9}\cdot \frac{4}{5}$

Simplify.

$\frac{4}{5}\cdot \left(\frac{5}{4}x\right)=\frac{{}^{2}\overline{)10}}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}9}\cdot \frac{4}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{)5}}_{1}}$

$x=\frac{8}{9}$

Therefore, the solution is $\frac{8}{9}$ .

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