202305111603 Exercise 3.1

Solve the following equations:

(a) $13x-4=3x+16$ .

(b) $\displaystyle{\frac{3x-9}{18}+\frac{x}{27}-\frac{2x-5}{4}=\frac{4}{3}-x}$ .

(c) $\displaystyle{\frac{3}{x-1}-\frac{2}{x+4}=\frac{4}{2-2x}}$ .

(d) $\displaystyle{\frac{2x}{x-3}-\frac{4x+1}{2x-1}=\frac{21}{2x^2-7x+3}}$ .

(e) $\displaystyle{\frac{1}{4}(3y-2)-\bigg[ y-\frac{1}{y}(7-3y)\bigg] =-\frac{1}{4}y-7}$ .

_{Extracted from K. L. Nielsen. (1958). College Mathematics.}

Roughwork.

(a)

$\begin{aligned} 13x-4 & = 3x+16 \\ 13x-3x & = 16+4 \\ 10x & = 20 \\ x & = 2 \\ \end{aligned}$

(b)

$\begin{aligned} \frac{3x-9}{18} + \frac{x}{27} - \frac{2x-5}{4} & = \frac{4}{3}-x \\ \frac{3x-9}{2\cdot 3^2} + \frac{x}{3^3} - \frac{2x-5}{2^2} & = \frac{4}{3}-x \\ 6(3x-9) + 4x-27(2x-5) & =36(4)-108x \\ 18x-54+4x-54x+135 & =144-108x \\ 18x+4x-54x+108x & = 144+54-135 \\ 76x & = 63 \\ x & = \frac{63}{76} \\ \end{aligned}$

(c)

$\begin{aligned} \frac{3}{x-1}-\frac{2}{x+4} & = \frac{4}{2-2x} \\ \frac{3}{x-1}-\frac{2}{x+4} & = -\frac{2}{x-1} \\ 3(x+4) - 2(x-1) & = -2(x+4) \\ 3x+12-2x+2 & = -2x-8 \\ 3x & = -22 \\ x & = -\frac{22}{3} \\ \end{aligned}$

(d) Not to be attempted.

(e) Not to be attempted.