Question 148094

Start with the given system of equations:

{{{system(x+y=-12,x-y=-4)}}}



Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(x+y)+(x-y)=(-12)+(-4)}}}



{{{(x+x)+(y-y)=-12+-4}}} Group like terms.



{{{2x+0y=-16}}} Combine like terms. Notice how the y terms cancel out.



{{{2x=-16}}} Simplify.



{{{x=(-16)/(2)}}} Divide both sides by {{{2}}} to isolate {{{x}}}.



{{{x=-8}}} Reduce.



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{{{x+y=-12}}} Now go back to the first equation.



{{{-8+y=-12}}} Plug in {{{x=-8}}}.



{{{-8+y=-12}}} Multiply.



{{{y=-12+8}}} Add {{{8}}} to both sides.



{{{y=-4}}} Combine like terms on the right side.



So our answer is {{{x=-8}}} and {{{y=-4}}}.



Which form the ordered pair *[Tex \LARGE \left(-8,-4\right)].



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(-8,-4\right)]. So this visually verifies our answer.



{{{drawing(500,500,-18,2,-14,6,
grid(1),
graph(500,500,-18,2,-14,6,-12-x,(-4-x)/(-1)),
circle(-8,-4,0.05),
circle(-8,-4,0.08),
circle(-8,-4,0.10)
)}}} Graph of {{{x+y=-12}}} (red) and {{{x-y=-4}}} (green)