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How do you solve the following system?: $- 13 x - 3 y = 6, 8 x - y = - 1$ $-13x -3y =6, 8x -y = -1$

1 Answer

Nov 19, 2015

$(x, y) = (- \frac{9}{37}, \frac{35}{37})$ $(x,y) = (-9/37,35/37)$
(see below for method of solution)

Explanation:

Given
[1] $XXX - 13 x - 3 y = 6$ $color(white)("XXX")-13x-3y=6$
[2] $XXX 8 x - y = - 1$ $color(white)("XXX")8x-y=-1$

Multiply [2] by 3 (to give $y$ $y$ the same coefficient as in [1])
[3] $XXX 24 x - 3 y = - 3$ $color(white)("XXX")24x-3y=-3$

Subtract [1] from [3] (to get rid of the $y$ $y$ term)
[4] $XXX 37 x = - 9$ $color(white)("XXX")37x=-9$

Divide [4] by 27
[5] $XXX x = - \frac{9}{37}$ $color(white)("XXX")x=-9/37$

Re-write [2] as an equation for $y$ $y$ by adding $(y + 1)$ $(y+1)$ to both sides and switching the sides
[6] $XXX y = 8 x + 1$ $color(white)("XXX")y=8x+1$

Substitute $(\frac{9}{37})$ $(9/37)$ from [5] for $x$ $x$ in [6]
[7] $XXX y = 8 (- \frac{9}{37}) + 1$ $color(white)("XXX")y=8(-9/37)+1$

Simplify [7]:
[8] $XXX y = - \frac{35}{37}$ $color(white)("XXX")y=-35/37$

Probably the biggest problem here is trusting your calculations (or being willing to do them in the first place) since the answer isn't "pretty".

If it helps a graph can help verify that the answer is within an appropriate range:
graph{(-13x-3y-6)*(8x-y+1)=0 [-3.97, 3.826, -2.976, 0.917]}

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