How do you solve the following system?: x +3y =8 , 6x +4y = -3

May 12, 2018

The values $x = - 2 \frac{13}{14}$ , $y = 3 \frac{9}{14}$ fulfills both equations.

Explanation:

It is always helpful to start with drawing the two equations as graphs in a diagram, not least as a check that our solution makes sense: We want to find a value of x and y which fulfill both equations at the same time, i.e. it must be on both lines f and g on the figure above.

As $x + 3 y = 8$, it follows that
(1) $x = - 3 y + 8$

Insert this value for x in the second equation. To make the solution simpler to follow I will write it as $- 3 = 6 x + 4 y$, that is:
$- 3 = 6 x + 4 y = 6 \left(- 3 y + 8\right) + 4 y$
$- 3 = - 18 y + 48 + 4 y = - 14 y + 48$

The last expression can be written as
$14 y = 48 + 3 = 51$ or $y = \frac{51}{14} = 3 \frac{9}{14}$

Insert this in expression (1):
$x = - 3 y + 8 = \left(- 3\right) \frac{51}{14} + 8 = \frac{- 3 \cdot 51 \cdot 8 \cdot 14}{14}$
$x = \frac{- 153 + 112}{14} = - \frac{41}{14} = - 2 \frac{13}{14}$

Our solution, therefore, is
$x = - 2 \frac{13}{14}$ , $y = 3 \frac{9}{14}$

If we convert these two values to decimal numbers, we see that it agrees with our graph.