Question 244763
# 1


Start with the given system of equations:

{{{system(-4x-2y=3,2x+y=1)}}}



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



{{{4x+2y=2}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-4x-2y=3,4x+2y=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:



{{{(-4x-2y)+(4x+2y)=(3)+(2)}}}



{{{(-4x+4x)+(-2y+2y)=3+2}}} Group like terms.



{{{0x+0y=5}}} Combine like terms.



{{{0=5}}}Simplify.



Since {{{0=5}}} is <font size="4"><b>NEVER</b></font> true, this means that there are no solutions. 



So the system is inconsistent.



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# 2



Start with the given system of equations:

{{{system(2x+y=1,-x-y=-3)}}}



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)=(1)+(-3)}}}



{{{(2x-x)+(y-y)=1-3}}} Group like terms.



{{{x+0y=-2}}} Combine like terms.



{{{x=-2}}} Simplify.



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



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



{{{-4+y=1}}} Multiply.



{{{y=1+4}}} Add {{{4}}} to both sides.



{{{y=5}}} Combine like terms on the right side.



So the solutions are {{{x=-2}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(-2,5\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(-2,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-12,8,-5,15,
grid(1),
graph(500,500,-12,8,-5,15,1-2x,(-3+x)/(-1)),
circle(-2,5,0.05),
circle(-2,5,0.08),
circle(-2,5,0.10)
)}}} Graph of {{{2x+y=1}}} (red) and {{{-x-y=-3}}} (green)