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How do you solve the following system?: $- 2 y = 2 x + 6, y = \frac{1}{2} x + 2$ $-2y=2x+6 , y= 1/2x +2$

1 Answer

May 13, 2017

$(x, y) = (- 3 \frac{1}{3}, \frac{1}{3})$ $(x,y)=(-3 1/3, 1/3)$

Explanation:

Given
[1] $XXX - 2 y = 2 x + 6$ $color(white)("XXX")-2y=2x+6$
[2] $XXX y = \frac{1}{2} x + 2$ $color(white)("XXX")y=1/2x+2$

Multiplying both sides of [2] by $(- 2)$ $(-2)$
[3] $XXX - 2 y = - x - 4$ $color(white)("XXX")-2y=-x-4$

Since the left sides of [1] and [3] are equal
the right sides must also be equal
[4] $XXX 2 x + 6 = - x - 4$ $color(white)("XXX")2x+6=-x-4$

Adding $x$ $x$ to both sides of [4]
[5] $XXX 3 x + 6 = - 4$ $color(white)("XXX")3x+6=-4$

Subtracting $6$ $6$ from both sides of [5]
[6] $XXX 3 x = - 10$ $color(white)("XXX")3x=-10$

Dividing both sides of [6] by $3$ $3$
[7] $XXX x = - \frac{10}{3} (= - 3 \frac{1}{3})$ $color(white)("XXX")x=-10/3 (=-3 1/3)$

Substituting $(- \frac{10}{3})$ $(-10/3)$ for $x$ $x$ in [2]
[8] $XXX y = \frac{1}{2} \times (- \frac{10}{3}) + 2$ $color(white)("XXX")y=1/2xx(-10/3)+2$

$XXXXX = - \frac{5}{3} + 2$ $color(white)("XXXXX")=-5/3+2$

$XXXXX = \frac{1}{3}$ $color(white)("XXXXX")=1/3$

$(x, y) = (- \frac{10}{3}, \frac{1}{3})$ $(x,y)=(-10/3,1/3)$

The graphic solution (below) supports this result:
enter image source here

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