# How do you solve the following linear system: x - 3y = 4x - 13 , y = 1/2x - 3/2 ?

Feb 26, 2017

$x = 3.89 \text{ , } y = 0.45$

#### Explanation:

$x - 3 y = 4 x - 13 \text{ (1)}$

$\textcolor{red}{y} = \frac{1}{2} x - \frac{3}{2} \text{ (2)}$

$\text{rearrange the equation (1) above}$

$x - 4 x = 3 y - 13$

$- 3 x = 3 y - 13$

$\text{divide both sides by 3}$

$\frac{- \cancel{3} x}{\cancel{3}} = \frac{3 y - 13}{3}$

$- x = \frac{\cancel{3} y}{\cancel{3}} - \frac{13}{3}$

$- x = y - \frac{13}{3}$

$\textcolor{red}{y} = - x + \frac{13}{3} \text{ (3)}$

$\text{the equations (3) and (2) are equal}$

$\text{we can write as ;}$

$\frac{1}{2} x - \frac{3}{2} = - x + \frac{13}{3}$

$\frac{1}{2} x + x = \frac{13}{3} + \frac{3}{2}$

$\frac{x}{2} + \textcolor{g r e e n}{\frac{2}{2}} x = \textcolor{g r e e n}{\frac{2}{2}} \cdot \frac{13}{3} + \textcolor{g r e e n}{\frac{3}{3}} \frac{3}{2}$

$\frac{3 x}{2} = \frac{26}{6} + \frac{9}{6}$

$\frac{3 x}{2} = \frac{35}{6}$

$\text{if "a/b=c/d" then } a \cdot d = b \cdot c$

$3 x \cdot 6 = 2 \cdot 35$

$18 x = 70$

$x = \frac{70}{18}$

$x = 3.89$

$\text{use (2)}$

$y = \frac{1}{2} x - \frac{3}{2}$

$y = \frac{1}{2} \cdot 3.89 - \frac{3}{2}$

$y = \frac{3.89 - 3}{2}$

$y = \frac{0.89}{2}$

$y = 0.45$