Question

Add the following algebraic expressions:
(i) 3a²b, − 4a²b, 9a²b
(ii) $\frac{2}{3}a,\frac{3}{5}a,-\frac{6}{5}a$
(iii) 4xy² − 7x²y, 12x²y − 6xy², − 3x²y +5xy²
(iv) $\frac{3}{2}a-\frac{5}{4}b+\frac{2}{5}c,\frac{2}{3}a-\frac{7}{2}b+\frac{7}{2}c,\frac{5}{3}a+\frac{5}{2}b-\frac{5}{4}c$
(v) $\frac{11}{2}xy+\frac{12}{5}y+\frac{13}{7}x,-\frac{11}{2}y-\frac{12}{5}x-\frac{13}{7}xy$
(vi) $\frac{7}{2}{x}^3-\frac{1}{2}{x}^2+\frac{5}{3},\frac{3}{2}{x}^3+\frac{7}{4}{x}^2-x+\frac{1}{3},\frac{3}{2}{x}^2-\frac{5}{2}x-2$

Answer 1

(i) To add the like terms, we proceed as follows:

$$\begin{gathered} 3{\mathrm{a}}^2\mathrm{b}+\left(-4{\mathrm{a}}^2\mathrm{b}\right)+9{\mathrm{a}}^2\mathrm{b} \\ =3{\mathrm{a}}^2\mathrm{b}-4{\mathrm{a}}^2\mathrm{b}+9{\mathrm{a}}^2\mathrm{b} \\ =\left(3-4+9\right){\mathrm{a}}^2\mathrm{b}(\mathrm{Distributive}\mathrm{law}) \\ =8{\mathrm{a}}^2\mathrm{b} \end{gathered}$$

(ii) To add the like terms, we proceed as follows:

$$\begin{gathered} \frac{2}{3}\mathrm{a}+\frac{3}{5}\mathrm{a}+\left(-\frac{6}{5}\mathrm{a}\right) \\ =\frac{2}{3}\mathrm{a}+\frac{3}{5}\mathrm{a}-\frac{6}{5}\mathrm{a} \\ =\left(\frac{2}{3}+\frac{3}{5}-\frac{6}{5}\right)\mathrm{a}(\mathrm{Distributive}\mathrm{law}) \\ =\frac{1}{15}\mathrm{a} \end{gathered}$$

(iii) To add, we proceed as follows:

$$\begin{gathered} \left(4{\mathrm{xy}}^2-7{\mathrm{x}}^2\mathrm{y}\right)+\left(12{\mathrm{x}}^2\mathrm{y}\right)+\left(-6{\mathrm{xy}}^2\right)+\left(-3{\mathrm{x}}^2\mathrm{y}+5{\mathrm{xy}}^2\right) \\ =4{\mathrm{xy}}^2-7{\mathrm{x}}^2\mathrm{y}+12{\mathrm{x}}^2\mathrm{y}-6{\mathrm{xy}}^2-3{\mathrm{x}}^2\mathrm{y}+5{\mathrm{xy}}^2 \\ =4{\mathrm{xy}}^2-6{\mathrm{xy}}^2+5{\mathrm{xy}}^2-7{\mathrm{x}}^2\mathrm{y}+12{\mathrm{x}}^2\mathrm{y}-3{\mathrm{x}}^2\mathrm{y}(\mathrm{Collecting}\mathrm{like}\mathrm{terms}) \\ =3{\mathrm{xy}}^2+2{\mathrm{x}}^2\mathrm{y}(\mathrm{Combining}\mathrm{like}\mathrm{terms}) \end{gathered}$$

(iv) To add, we proceed as follows:

$$\begin{gathered} \left(\frac{3}{2}\mathrm{a}-\frac{5}{4}\mathrm{b}+\frac{2}{5}\mathrm{c}\right)+\left(\frac{2}{3}\mathrm{a}-\frac{7}{2}\mathrm{b}+\frac{7}{2}\mathrm{c}\right)+\left(\frac{5}{3}\mathrm{a}+\frac{5}{2}\mathrm{b}-\frac{5}{4}\mathrm{c}\right) \\ =\frac{3}{2}\mathrm{a}-\frac{5}{4}\mathrm{b}+\frac{2}{5}\mathrm{c}+\frac{2}{3}\mathrm{a}-\frac{7}{2}\mathrm{b}+\frac{7}{2}\mathrm{c}+\frac{5}{3}\mathrm{a}+\frac{5}{2}\mathrm{b}-\frac{5}{4}\mathrm{c} \\ =\frac{3}{2}\mathrm{a}+\frac{2}{3}\mathrm{a}+\frac{5}{3}\mathrm{a}-\frac{5}{4}\mathrm{b}-\frac{7}{2}\mathrm{b}+\frac{5}{2}\mathrm{b}+\frac{2}{5}\mathrm{c}+\frac{7}{2}\mathrm{c}-\frac{5}{4}\mathrm{c}(\mathrm{Collecting}\mathrm{like}\mathrm{terms}) \\ =\frac{23}{6}\mathrm{a}-\frac{9}{4}\mathrm{b}+\frac{53}{20}\mathrm{c}(\mathrm{Combining}\mathrm{like}\mathrm{terms}) \end{gathered}$$

(v) To add, we proceed as follows:

$$\begin{gathered} \left(\frac{11}{2}xy+\frac{12}{5}y+\frac{13}{7}x\right)+\left(-\frac{11}{2}y-\frac{12}{5}x-\frac{13}{7}xy\right) \\ =\frac{11}{2}xy+\frac{12}{5}y+\frac{13}{7}x-\frac{11}{2}y-\frac{12}{5}x-\frac{13}{7}xy \\ =\frac{11}{2}xy-\frac{13}{7}xy+\frac{12}{5}y-\frac{11}{2}y+\frac{13}{7}x-\frac{12}{5}x(\text{Collecting like terms)} \\ =\frac{51}{14}xy-\frac{31}{10}y-\frac{19}{35}x(\text{Combining like terms)} \end{gathered}$$

(vi) To add, we proceed as follows:

$$\begin{gathered} \left(\frac{7}{2}{x}^3-\frac{1}{2}{x}^2+\frac{5}{3}\right)+\left(\frac{3}{2}{x}^3+\frac{7}{4}{x}^2-x+\frac{1}{3}\right)+\left(\frac{3}{2}{x}^2-\frac{5}{2}x-2\right) \\ =\frac{7}{2}{x}^3-\frac{1}{2}{x}^2+\frac{5}{3}+\frac{3}{2}{x}^3+\frac{7}{4}{x}^2-x+\frac{1}{3}+\frac{3}{2}{x}^2-\frac{5}{2}x-2 \\ =\frac{7}{2}{x}^3+\frac{3}{2}{x}^3-\frac{1}{2}{x}^2+\frac{7}{4}{x}^2+\frac{3}{2}{x}^2-x-\frac{5}{2}x+\frac{5}{3}+\frac{1}{3}-2(\mathrm{Collecting}\mathrm{like}\mathrm{terms}) \\ =5{\mathrm{x}}^3+\frac{11}{4}{\mathrm{x}}^2-\frac{7}{2}\mathrm{x}(\mathrm{Combining}\mathrm{like}\mathrm{terms}) \end{gathered}$$