Question 402977


Start with the given system of equations:

{{{system(-7x-5y=-3,x-8y=-17)}}}



{{{7(x-8y)=7(-17)}}} Multiply the both sides of the second equation by 7.



{{{7x-56y=-119}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-7x-5y=-3,7x-56y=-119)}}}



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



{{{(-7x-5y)+(7x-56y)=(-3)+(-119)}}}



{{{(-7x+7x)+(-5y+-56y)=-3+-119}}} Group like terms.



{{{0x+-61y=-122}}} Combine like terms. Notice how the x terms cancel out.



{{{-61y=-122}}} Simplify.



{{{y=(-122)/(-61)}}} Divide both sides by {{{-61}}} to isolate {{{y}}}.



{{{y=2}}} Reduce.



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{{{-7x-5y=-3}}} Now go back to the first equation.



{{{-7x-5(2)=-3}}} Plug in {{{y=2}}}.



{{{-7x-10=-3}}} Multiply.



{{{-7x=-3+10}}} Add {{{10}}} to both sides.



{{{-7x=7}}} Combine like terms on the right side.



{{{x=(7)/(-7)}}} Divide both sides by {{{-7}}} to isolate {{{x}}}.



{{{x=-1}}} Reduce.



So our answer is {{{x=-1}}} and {{{y=2}}}.



Which form the ordered pair *[Tex \LARGE \left(-1,2\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(-1,2\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11,9,-8,12,
grid(1),
graph(500,500,-11,9,-8,12,(-3+7x)/(-5),(-17-x)/(-8)),
circle(-1,2,0.05),
circle(-1,2,0.08),
circle(-1,2,0.10)
)}}} Graph of {{{-7x-5y=-3}}} (red) and {{{x-8y=-17}}} (green)