Question 550476


Start with the given system of equations:

{{{system(7x+6y=-5,-7x+y=40)}}}



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



{{{(7x+6y)+(-7x+y)=(-5)+(40)}}}



{{{(7x+-7x)+(6y+1y)=-5+40}}} Group like terms.



{{{0x+7y=35}}} Combine like terms.



{{{7y=35}}} Simplify.



{{{y=(35)/(7)}}} Divide both sides by {{{7}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



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{{{7x+6y=-5}}} Now go back to the first equation.



{{{7x+6(5)=-5}}} Plug in {{{y=5}}}.



{{{7x+30=-5}}} Multiply.



{{{7x=-5-30}}} Subtract {{{30}}} from both sides.



{{{7x=-35}}} Combine like terms on the right side.



{{{x=(-35)/(7)}}} Divide both sides by {{{7}}} to isolate {{{x}}}.



{{{x=-5}}} Reduce.



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



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



{{{drawing(500,500,-15,5,-5,15,
grid(1),
graph(500,500,-15,5,-5,15,(-5-7x)/(6),40+7x),
circle(-5,5,0.05),
circle(-5,5,0.08),
circle(-5,5,0.10)
)}}} Graph of {{{7x+6y=-5}}} (red) and {{{-7x+y=40}}} (green) 



So again, the answer is the ordered pair (-5, 5)