Question

Add the following expressions:

(i) 5x, 7x, −6x
(ii) $\frac{3}{5}x,\frac{2}{3}x,\frac{-4}{5}x$
(iii) 5a²b, −8a²b, 7a²b
(iv) $\frac{3}{4}{x}^2,5x^2,-3x^2,-\frac{1}{4}{x}^2$
(v) x − 3y + 4z, y − 2x − 8z, 5x − 2y − 3z
(vi) 2x² − 3y², 5x² + 6y², − 3x² − 4y²
(vii) 5x − 2x² − 8, 8x² − 7x − 9, 3 + 7x² − 2x
(viii) $\frac{2}{3}a-\frac{4}{5}b+\frac{3}{5}c,-\frac{3}{4}a-\frac{5}{2}b+\frac{2}{3}c,\frac{5}{2}a+\frac{7}{4}b-\frac{5}{6}c$
(ix) $\frac{8}{5}x+\frac{11}{7}y+\frac{9}{4}xy,-\frac{3}{2}x-\frac{5}{3}y-\frac{9}{5}xy$
(x) $\frac{3}{2}{x}^3-\frac{1}{4}{x}^2+\frac{5}{3},-\frac{5}{4}{x}^3+\frac{3}{5}{x}^2-x+\frac{1}{5},-x^2+\frac{3}{8}x-\frac{8}{15}$

Answer 1

(i)
5x + 7x + (-6x)
= 5x + 7x -6x
= 6x

(ii)

$$\begin{gathered} \frac{3}{5}\mathrm{x}+\frac{2}{3}\mathrm{x}+\frac{-4}{5}\mathrm{x} \\ =\frac{9\mathrm{x}+10\mathrm{x}-12\mathrm{x}}{15}=\frac{7\mathrm{x}}{15} \end{gathered}$$

(iii)
5a²b +( −8a²b) + 7a²b
= 5a²b − 8a²b + 7a²b
4a²b

(iv)
$$\begin{gathered} \frac{3}{4}{\mathrm{x}}^2+5{\mathrm{x}}^2+(-3{\mathrm{x}}^2)+(-\frac{1}{4}{\mathrm{x}}^2) \\ =\frac{3}{4}{\mathrm{x}}^2-\frac{1}{4}{\mathrm{x}}^2+5{\mathrm{x}}^2-3{\mathrm{x}}^2 \\ =\frac{1}{2}{\mathrm{x}}^2+2{\mathrm{x}}^2=\frac{5}{2}{\mathrm{x}}^2 \end{gathered}$$


(v) Collecting like terms and adding them:

x − 3y + 4z + y − 2x − 8z + 5x − 2y − 3z
= x- 2x + 5x - 3y + y - 2y + 4z - 8z - 3z
= 4x -4y -7z

(vi) Collecting like terms and adding them:
2x² − 3y² + 5x² + 6y² + (− 3x² − 4y²)
$$\begin{gathered} =2{\mathrm{x}}^2+5{\mathrm{x}}^2-3{\mathrm{x}}^2-3{\mathrm{y}}^2+6{\mathrm{y}}^2-4{\mathrm{y}}^2 \\ =4{\mathrm{x}}^2-{\mathrm{y}}^2 \end{gathered}$$

(vii) Collecting like terms and adding them:
5x − 2x² − 8 + 8x² − 7x − 9 + 3 + 7x² − 2x
$$\begin{gathered} =-2{\mathrm{x}}^2+8{\mathrm{x}}^2+7{\mathrm{x}}^2+5\mathrm{x}-7\mathrm{x}-2\mathrm{x}-8-9+3 \\ =13{\mathrm{x}}^2-4\mathrm{x}-14 \end{gathered}$$


(viii) Collecting like terms and adding them:

$$\begin{gathered} \frac{2}{3}\mathrm{a}-\frac{4}{5}\mathrm{b}+\frac{3}{5}\mathrm{c}+\left(-\frac{3}{4}\mathrm{a}-\frac{5}{2}\mathrm{b}+\frac{2}{3}\mathrm{c}\right)+\frac{5}{2}\mathrm{a}+\frac{7}{4}\mathrm{b}-\frac{5}{6}\mathrm{c} \\ \mathrm{b}-\frac{5}{2}\mathrm{b}+\frac{7}{4}\mathrm{b}+\frac{3}{5}\mathrm{c}+\frac{2}{3}\mathrm{c}-\frac{5}{6}\mathrm{c} \\ =\frac{(8-9+30)\mathrm{a}}{12}+\frac{(-16-50+35)\mathrm{b}}{20}+\frac{(18+20-25)\mathrm{c}}{30} \\ =\frac{29}{12}\mathrm{a}-\frac{31}{20}\mathrm{b}+\frac{13}{30}\mathrm{c} \end{gathered}$$


(ix) Collecting like terms and adding them:
$$\begin{gathered} \frac{8}{5}\mathrm{x}+\frac{11}{7}\mathrm{y}+\frac{9}{4}\mathrm{xy}+\left(-\frac{3}{2}\mathrm{x}-\frac{5}{3}\mathrm{y}-\frac{9}{5}\mathrm{xy}\right) \\ =\frac{8}{5}\mathrm{x}-\frac{3}{2}\mathrm{x}+\frac{11}{7}\mathrm{y}-\frac{5}{3}\mathrm{y}+\frac{9}{4}\mathrm{xy}-\frac{9}{5}\mathrm{xy} \\ =\frac{1}{10}\mathrm{x}-\frac{2}{21}\mathrm{y}+\frac{9}{20}\mathrm{xy} \end{gathered}$$

(x) Collecting like terms and adding them:
$$\begin{gathered} \frac{3}{2}{\mathrm{x}}^3-\frac{1}{4}{\mathrm{x}}^2+\frac{5}{3}+\left(-\frac{5}{4}{\mathrm{x}}^3+\frac{3}{5}{\mathrm{x}}^2-\mathrm{x}+\frac{1}{5}\right)+\left(-{\mathrm{x}}^2+\frac{3}{8}\mathrm{x}-\frac{8}{15}\right) \\ =\frac{3}{2}{\mathrm{x}}^3-\frac{5}{4}{\mathrm{x}}^3-\frac{1}{4}{\mathrm{x}}^2+\frac{3}{5}{\mathrm{x}}^2-{\mathrm{x}}^2-\mathrm{x}+\frac{3}{8}\mathrm{x}+\frac{5}{3}+\frac{1}{5}-\frac{8}{15} \\ =\frac{1}{4}{\mathrm{x}}^3-\frac{13}{20}{\mathrm{x}}^2-\frac{5}{8}\mathrm{x}+\frac{4}{3} \end{gathered}$$x+117y+94xy,−32x−53y−9