Question 183268

Start with the given system of equations:

{{{system(4x-y=8,2x+y=4)}}}



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



{{{(4x-y)+(2x+y)=(8)+(4)}}}



{{{(4x+2x)+(-y+y)=8+4}}} Group like terms.



{{{6x+0y=12}}} Combine like terms.



{{{6x=12}}} Simplify.



{{{x=(12)/(6)}}} Divide both sides by {{{6}}} to isolate {{{x}}}.



{{{x=2}}} Reduce.



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{{{4x-y=8}}} Now go back to the first equation.



{{{4(2)-y=8}}} Plug in {{{x=2}}}.



{{{8-y=8}}} Multiply.



{{{-y=8-8}}} Subtract {{{8}}} from both sides.



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



{{{y=(0)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{y}}}.



{{{y=0}}} Reduce.



So the solutions are {{{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,(8-4x)/(-1),4-2x),
circle(2,0,0.05),
circle(2,0,0.08),
circle(2,0,0.10)
)}}} Graph of {{{4x-y=8}}} (red) and {{{2x+y=4}}} (green)