How do solve the following linear system?:  3/2x - 3y = -9/4 , 2/3x + 4y = -1 ?

1 Answer
Feb 19, 2016

$\left(x , y\right) = \left(- \frac{3}{2} , 0\right)$

Explanation:

Given:
[1]$\textcolor{w h i t e}{\text{XXX}} \frac{3}{2} x - 3 y = - \frac{9}{4}$
[2]$\textcolor{w h i t e}{\text{XXX}} \frac{2}{3} x + 4 y = - 1$
To make the solution easier we will clear the fractions by
multiplying [1] by $4$ and [2] by $3$ to get
[3]$\textcolor{w h i t e}{\text{XXX}} 6 x - 12 y = - 9$
[4]$\textcolor{w h i t e}{\text{XXX}} 2 x + 12 y = - 3$

Adding equations [3] and [4] gives
[5]$\textcolor{w h i t e}{\text{XXX}} 8 x = - 12$
and after dividing by $8$
[6]$\textcolor{w h i t e}{\text{XXX}} x = - \frac{3}{2}$

Substituting $\left(- \frac{3}{2}\right)$ (from [6]) for $x$ in [2]
[7]$\textcolor{w h i t e}{\text{XXX}} \frac{2}{3} \times \left(- \frac{3}{2}\right) + 4 y = - 1$
Simplifying
[8]$\textcolor{w h i t e}{\text{XXX}} y = 0$

Graphically:
graph{(3/2x-3y+9/4)(2/3x+4y+1)=0 [-5.07, 2.725, -1.414, 2.483]}