Question 590778


Start with the given system of equations:

{{{system(-7x-y=-30,4x+y=21)}}}



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



{{{(-7x-y)+(4x+y)=(-30)+(21)}}}



{{{(-7x+4x)+(-y+y)=-30+21}}} Group like terms.



{{{-3x+0y=-9}}} Combine like terms.



{{{-3x=-9}}} Simplify.



{{{x=(-9)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{x}}}.



{{{x=3}}} Reduce.



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{{{-7x-y=-30}}} Now go back to the first equation.



{{{-7(3)-y=-30}}} Plug in {{{x=3}}}.



{{{-21-y=-30}}} Multiply.



{{{-y=-30+21}}} Add {{{21}}} to both sides.



{{{-y=-9}}} Combine like terms on the right side.



{{{y=(-9)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{y}}}.



{{{y=9}}} Reduce.



So the solutions are {{{x=3}}} and {{{y=9}}}.



Which form the ordered pair *[Tex \LARGE \left(3,9\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(3,9\right)]. So this visually verifies our answer.



{{{drawing(500,500,-7,13,-1,19,
grid(1),
graph(500,500,-7,13,-1,19,(-30+7x)/(-1),21-4x),
circle(3,9,0.05),
circle(3,9,0.08),
circle(3,9,0.10)
)}}} Graph of {{{-7x-y=-30}}} (red) and {{{4x+y=21}}} (green) 



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