Question 169997
{{{y=-x+2}}} Start with the first equation



{{{x+y=2}}} Add "x" to both sides





So we have the system of equations:



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



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



{{{-1x-1y=-2}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-1x-1y=-2,x+2y=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:



{{{(-1x-1y)+(x+2y)=(-2)+(6)}}}



{{{(-1x+1x)+(-1y+2y)=-2+6}}} Group like terms.



{{{0x+y=4}}} Combine like terms. Notice how the x terms cancel out.



{{{y=4}}} Simplify.



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{{{-1x-1y=-2}}} Now go back to the first equation.



{{{-1x-1(4)=-2}}} Plug in {{{y=4}}}.



{{{-1x-4=-2}}} Multiply.



{{{-x=-2+4}}} Add {{{4}}} to both sides.



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



{{{x=(2)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{x}}}.



{{{x=-2}}} Reduce.



So our answer is {{{x=-2}}} and {{{y=4}}}.



Which form the ordered pair *[Tex \LARGE \left(-2,4\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(-2,4\right)]. So this visually verifies our answer.



{{{drawing(500,500,-12,8,-6,14,
grid(1),
graph(500,500,-12,8,-6,14,2-x,(6-x)/(2)),
circle(-2,4,0.05),
circle(-2,4,0.08),
circle(-2,4,0.10)
)}}} Graph of {{{y=-x+2}}} (red) and {{{x+2y=6}}} (green)