Question 154416

Start with the given system of equations:

{{{system(2x+y=4,x+y=2)}}}



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



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



So we have the new system of equations:

{{{system(2x+y=4,-x-y=-2)}}}



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+y)+(-x-y)=(4)+(-2)}}}



{{{(2x-x)+(y-y)=4+-2}}} Group like terms.



{{{x+0y=2}}} Combine like terms. Notice how the y terms cancel out.



{{{x=2}}} Simplify.



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{{{2x+y=4}}} Now go back to the first equation.



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



{{{4+y=4}}} Multiply.



{{{y=4-4}}} Subtract {{{4}}} from both sides.



{{{y=0}}} Combine like terms on the right side.



So our answer is {{{x=2}}} and {{{y=0}}}.



Which form the ordered pair *[Tex \LARGE \left(2,0\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(2,0\right)]. So this visually verifies our answer.



{{{drawing(500,500,-8,12,-10,10,
grid(1),
graph(500,500,-8,12,-10,10,4-2x,2-x),
circle(2,0,0.05),
circle(2,0,0.08),
circle(2,0,0.10)
)}}} Graph of {{{2x+y=4}}} (red) and {{{x+y=2}}} (green)