5.3Fractional Equations

A fractional equation has the unknown in the denominator. This is different from an equation with fractional coefficients (unknown in the numerator).

$\dfrac{x + 2}{3} = 5$ , fractional coefficient (unknown in numerator)
$\dfrac{9}{2x - 5} = 3$ , fractional equation (unknown in denominator)

Unknown in denominator

Solve $\displaystyle \frac{20}{3x + 1} = 4$ .

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\displaystyle 20

\displaystyle = 4(3x + 1) \quad \scriptsize\textit{multiply both sides by (3x + 1)}

\displaystyle 20

\displaystyle = 12x + 4 \quad \scriptsize\textit{expand}

\displaystyle 12x

\displaystyle = 16

\displaystyle x

\displaystyle = \frac{4}{3}

Cross multiplication

Solve $\displaystyle \frac{n - 3}{n + 2} = \frac{3}{7}$ .

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\displaystyle 7(n - 3)

\displaystyle = 3(n + 2) \quad \scriptsize\textit{cross multiply}

\displaystyle 7n - 21

\displaystyle = 3n + 6 \quad \scriptsize\textit{expand}

\displaystyle 4n

\displaystyle = 27

\displaystyle n

\displaystyle = \frac{27}{4}

Now you try

Solve $\frac{2}{y - 5} = \frac{3}{y - 6}$ .

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\displaystyle \frac{2}{y - 5}

\displaystyle = \frac{3}{y - 6}

\displaystyle 2(y - 6)

\displaystyle = 3(y - 5) \quad \scriptsize\textit{cross multiply}

\displaystyle 2y - 12

\displaystyle = 3y - 15

\displaystyle -y

\displaystyle = -3

\displaystyle y

\displaystyle = 3

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