Question 374764


Start with the given system of equations:

{{{system(-8x-7y=2,7x+6y=-1)}}}



{{{6(-8x-7y)=6(2)}}} Multiply the both sides of the first equation by 6.



{{{-48x-42y=12}}} Distribute and multiply.



{{{7(7x+6y)=7(-1)}}} Multiply the both sides of the second equation by 7.



{{{49x+42y=-7}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-48x-42y=12,49x+42y=-7)}}}



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



{{{(-48x-42y)+(49x+42y)=(12)+(-7)}}}



{{{(-48x+49x)+(-42y+42y)=12+-7}}} Group like terms.



{{{x+0y=5}}} Combine like terms.



{{{x=5}}} Simplify.



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{{{-48x-42y=12}}} Now go back to the first equation.



{{{-48(5)-42y=12}}} Plug in {{{x=5}}}.



{{{-240-42y=12}}} Multiply.



{{{-42y=12+240}}} Add {{{240}}} to both sides.



{{{-42y=252}}} Combine like terms on the right side.



{{{y=(252)/(-42)}}} Divide both sides by {{{-42}}} to isolate {{{y}}}.



{{{y=-6}}} Reduce.



So the solutions are {{{x=5}}} and {{{y=-6}}}.



Which form the ordered pair *[Tex \LARGE \left(5,-6\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(5,-6\right)]. So this visually verifies our answer.



{{{drawing(500,500,-5,15,-16,4,
grid(1),
graph(500,500,-5,15,-16,4,(2+8x)/(-7),(-1-7x)/(6)),
circle(5,-6,0.05),
circle(5,-6,0.08),
circle(5,-6,0.10)
)}}} Graph of {{{-8x-7y=2}}} (red) and {{{7x+6y=-1}}} (green)