# Solving One-Step Linear Equations with Mixed Numbers

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.

Some linear equations can be solved with a single operation. For this type of equation, use the inverse operation to solve. The easiest type involves only an addition or a subtraction.

**
Example 1:
**

Solve.

$\frac{3}{4}+p=1\frac{1}{4}$

Rewrite the mixed number as an improper fraction.

$\frac{3}{4}+p=\frac{5}{4}$

The inverse operation of addition is subtraction. Use the subtraction property of equality to subtract $\frac{3}{4}$ from both sides.

$\frac{3}{4}+p-\frac{3}{4}=\frac{5}{4}-\frac{3}{4}$

Simplify.

$p=\frac{2}{4}$

Divide the numerator and the denominator by the GCF, $2$ .

$p=\frac{1}{2}$

Also, 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 2:
**

Solve.
*
*

$2\frac{1}{4}x=7\frac{1}{6}$

Rewrite the mixed numbers as improper fractions.

$\frac{9}{4}x=\frac{43}{6}$

The inverse operation of division is multiplication.

To isolate the variable $x$ (to get a coefficient of $1$ ), multiply both sides by $\frac{4}{9}$ which is the reciprocal of $\frac{9}{4}$ .

$\frac{4}{9}\cdot \left(\frac{9}{4}x\right)=\frac{4}{9}\cdot \frac{43}{6}$

You can cancel common factors before you multiply.

$\begin{array}{l}x=\frac{{}^{2}\overline{)4}}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}9}\cdot \frac{43}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{)6}}_{2}}\hfill \\ x=\frac{86}{27}\hfill \end{array}$

Finally, write the improper fraction as a mixed number. $27$ goes into $86$ three times with a remainder of $5$ . So:

$x=3\frac{5}{27}$