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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 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.

3 4 + p = 1 1 4

Rewrite the mixed number as an improper fraction.

3 4 + p = 5 4

The inverse operation of addition is subtraction. Use the subtraction property of equality to subtract 3 4 from both sides.

3 4 + p 3 4 = 5 4 3 4

Simplify.

p = 2 4

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

p = 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 1 4 x = 7 1 6

Rewrite the mixed numbers as improper fractions.

9 4 x = 43 6

The inverse operation of division is multiplication.

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

4 9 ( 9 4 x ) = 4 9 43 6

You can cancel common factors before you multiply.

x = 4 2 9 43 6 2 x = 86 27

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

x = 3 5 27