Question

Express the following difference in the simplest form $?$

$\frac{3}{\mathrm{d}-7}-\frac{2}{3\mathrm{d}+1}$

Answer 1

Find the difference in the simplest form:

$$\begin{gathered} \frac{3}{\mathrm{d}-7}-\frac{2}{3\mathrm{d}+1}=\frac{3(3\mathrm{d}+1)-2(\mathrm{d}-7)}{(\mathrm{d}-7)(3\mathrm{d}+1)}(\mathrm{simplify}\mathrm{by}\mathrm{taking}\mathrm{least}\mathrm{common}\mathrm{denominator}) \\ =\frac{9\mathrm{d}+3-2\mathrm{d}+14}{(\mathrm{d}-7)(3\mathrm{d}+1)} \end{gathered}$$

(group the like terms together)

$\Rightarrow \frac{3}{\mathrm{d}-7}-\frac{2}{3\mathrm{d}+1}=\frac{(9\mathrm{d}-2\mathrm{d})+(3+14)}{(\mathrm{d}-7)(3\mathrm{d}+1)}$ (combine the like terms)

$\Rightarrow \frac{3}{\mathrm{d}-7}-\frac{2}{3\mathrm{d}+1}=\frac{7\mathrm{d}+17}{(\mathrm{d}-7)(3\mathrm{d}+1)}$

Hence, the difference in the simplest form is $\frac{\mathbf{7}\mathbf{d}\mathbf{+}\mathbf{17}}{\mathbf{(}\mathbf{d}\mathbf{-}\mathbf{7}\mathbf{)}\mathbf{(}\mathbf{3}\mathbf{d}\mathbf{+}\mathbf{1}\mathbf{)}}$.