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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 $\frac{9}{4}$ .

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

You can cancel common factors before you multiply.

$\begin{array}{l} x = \frac{{}^{2}4}{9} \cdot \frac{43}{6_{2}} \\ x = \frac{86}{27} \end{array}$

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

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