Question 157504


Start with the given system of equations:

{{{system(6x-y=49,x+6y=39)}}}



{{{-6(x+6y)=-6(39)}}} Multiply the both sides of the second equation by -6.



{{{-6x-36y=-234}}} Distribute and multiply.



So we have the new system of equations:

{{{system(6x-1y=49,-6x-36y=-234)}}}



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



{{{(6x-1y)+(-6x-36y)=(49)+(-234)}}}



{{{(6x+-6x)+(-1y+-36y)=49+-234}}} Group like terms.



{{{0x+-37y=-185}}} Combine like terms. Notice how the x terms cancel out.



{{{-37y=-185}}} Simplify.



{{{y=(-185)/(-37)}}} Divide both sides by {{{-37}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



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{{{6x-1y=49}}} Now go back to the first equation.



{{{6x-1(5)=49}}} Plug in {{{y=5}}}.



{{{6x-5=49}}} Multiply.



{{{6x=49+5}}} Add {{{5}}} to both sides.



{{{6x=54}}} Combine like terms on the right side.



{{{x=(54)/(6)}}} Divide both sides by {{{6}}} to isolate {{{x}}}.



{{{x=9}}} Reduce.



So our answer is {{{x=9}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(9,5\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(9,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-1,19,-5,15,
grid(1),
graph(500,500,-1,19,-5,15,(49-6x)/(-1),(39-x)/(6)),
circle(9,5,0.05),
circle(9,5,0.08),
circle(9,5,0.10)
)}}} Graph of {{{6x-y=49}}} (red) and {{{x+6y=39}}} (green) 



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