Question 157505


Start with the given system of equations:

{{{system(6x-4y=62,3x+6y=-9)}}}



{{{-2(3x+6y)=-2(-9)}}} Multiply the both sides of the second equation by -2.



{{{-6x-12y=18}}} Distribute and multiply.



So we have the new system of equations:

{{{system(6x-4y=62,-6x-12y=18)}}}



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-4y)+(-6x-12y)=(62)+(18)}}}



{{{(6x+-6x)+(-4y+-12y)=62+18}}} Group like terms.



{{{0x+-16y=80}}} Combine like terms. Notice how the x terms cancel out.



{{{-16y=80}}} Simplify.



{{{y=(80)/(-16)}}} Divide both sides by {{{-16}}} to isolate {{{y}}}.



{{{y=-5}}} Reduce.



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{{{6x-4y=62}}} Now go back to the first equation.



{{{6x-4(-5)=62}}} Plug in {{{y=-5}}}.



{{{6x+20=62}}} Multiply.



{{{6x=62-20}}} Subtract {{{20}}} from both sides.



{{{6x=42}}} Combine like terms on the right side.



{{{x=(42)/(6)}}} Divide both sides by {{{6}}} to isolate {{{x}}}.



{{{x=7}}} Reduce.



So our answer is {{{x=7}}} and {{{y=-5}}}.



Which form the ordered pair *[Tex \LARGE \left(7,-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(7,-5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-3,17,-15,5,
grid(1),
graph(500,500,-3,17,-15,5,(62-6x)/(-4),(-9-3x)/(6)),
circle(7,-5,0.05),
circle(7,-5,0.08),
circle(7,-5,0.10)
)}}} Graph of {{{6x-4y=62}}} (red) and {{{3x+6y=-9}}} (green) 



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