Question 147493

{{{-9=x-3y}}} Start with the first equation.



{{{0=x-3y+9}}} Add {{{9}}} to both sides.



{{{-x=-3y+9}}} Subtract {{{x}}} from both sides.



{{{-x+3y=9}}} Add {{{3y}}} to both sides.



----------------------------


{{{-2x+6=6y}}} Start with the second equation.



{{{-2x=6y-6}}} Subtract {{{6}}} from both sides.



{{{-2x-6y=-6}}} Subtract {{{6y}}} from both sides.



So we have the system of equations:



{{{system(-x+3y=9,-2x-6y=-6)}}}



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



{{{-2x+6y=18}}} Distribute and multiply.



{{{-1(-2x-6y)=-1(-6)}}} Multiply the both sides of the second equation by -1.



{{{2x+6y=6}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-2x+6y=18,2x+6y=6)}}}



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



{{{(-2x+6y)+(2x+6y)=(18)+(6)}}}



{{{(-2x+2x)+(6y+6y)=18+6}}} Group like terms.



{{{0x+12y=24}}} Combine like terms. Notice how the x terms cancel out.



{{{12y=24}}} Simplify.



{{{y=(24)/(12)}}} Divide both sides by {{{12}}} to isolate {{{y}}}.



{{{y=2}}} Reduce.



------------------------------------------------------------------



{{{-2x+6y=18}}} Now go back to the first equation.



{{{-2x+6(2)=18}}} Plug in {{{y=2}}}.



{{{-2x+12=18}}} Multiply.



{{{-2x=18-12}}} Subtract {{{12}}} from both sides.



{{{-2x=6}}} Combine like terms on the right side.



{{{x=(6)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{x}}}.



{{{x=-3}}} Reduce.



So our answer is {{{x=-3}}} and {{{y=2}}}.



Which form the ordered pair *[Tex \LARGE \left(-3,2\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(-3,2\right)]. So this visually verifies our answer.



{{{drawing(500,500,-13,7,-8,12,
grid(1),
graph(500,500,-13,7,-8,12,(9+x)/(3),(-6+2x)/(-6)),
circle(-3,2,0.05),
circle(-3,2,0.08),
circle(-3,2,0.10)
)}}} Graph of {{{-x+3y=9}}} (red) and {{{-2x-6y=-6}}} (green)