How to solve equations with fractions
What is a fractional equation?
In this lesson we’ll look at how to solve equations with numerical fractions as coefficients and terms.
Remember that multiplying a fraction by its reciprocal will always give you a value of ???1???.
For example ???4/5??? has a reciprocal of ???5/4??? because
???\frac{4}{5}\cdot\frac{5}{4}=1???
To clear a fraction from an equation, multiply all of the terms on both sides of the equation by the fraction’s denominator.
For example, to clear the ???2??? from the fraction in ???5x+1/2=12???, multiply the equation by ???2??? on both sides.
???2\left(5x+\frac{1}{2}=12\right)???
???2(5x)+2\left(\frac{1}{2}\right)=2(12)???
???10x+1=24???
How to solve equations when there’s a fraction somewhere in the equation
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Clearing the fraction from the equation in order to solve for the variable
Example
Solve for the variable.
???\frac{4}{5}n=20???
To get rid of a fractional coefficient, we have to multiply both sides by its reciprocal, because that’ll make the fraction ???1???.
???\frac{4}{5}n=20???
???\frac{5}{4}\cdot\frac{4}{5}n=\frac{5}{4}\cdot20???
???\frac{20}{20}n=\frac{100}{4}???
???1n=25???
???n=25???
If you have a fractional coefficient and another term, you can isolate the term with the variable and then multiply both sides by the reciprocal of the fractional coefficient.
Example
Solve for the variable.
???\frac{4}{7}x+14=22???
First isolate the fractional term.
???\frac{4}{7}x+14-14=22-14???
???\frac{4}{7}x=8???
Now get rid of the fractional coefficient by multiplying both sides of the equation by the reciprocal of ???4/7???.
???\frac{7}{4}\cdot\frac{4}{7}x=\frac{7}{4}\cdot8???
???\frac{28}{28}x=\frac{56}{4}???
???1x=14???
???x=14???
We could also do this same problem by first clearing the fraction. In order to get rid of the fraction, we have to multiply every term in the equation by its denominator.
???7\left(\frac{4}{7}x+14=22\right)???
???7\cdot\frac{4}{7}x+7\cdot14=7\cdot22???
???4x+98=154???
Now we can solve for the variable using inverse operations.
???4x+98-98=154-98???
???4x=56???
???\frac{4x}{4}=\frac{56}{4}???
???x=14???