Question 200121


Start with the given system of equations:

{{{system(-x+4y=-11,2x+4y=-2)}}}



{{{2(-x+4y)=2(-11)}}} Multiply the both sides of the first equation by 2.



{{{-2x+8y=-22}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-2x+8y=-22,2x+4y=-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+8y)+(2x+4y)=(-22)+(-2)}}}



{{{(-2x+2x)+(8y+4y)=-22+-2}}} Group like terms.



{{{0x+12y=-24}}} Combine like terms.



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



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



{{{y=-2}}} Reduce.



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{{{-2x+8y=-22}}} Now go back to the first equation.



{{{-2x+8(-2)=-22}}} Plug in {{{y=-2}}}.



{{{-2x-16=-22}}} Multiply.



{{{-2x=-22+16}}} Add {{{16}}} to 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 the solutions are {{{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,-7,13,-12,8,
grid(1),
graph(500,500,-7,13,-12,8,(-11+x)/(4),(-2-2x)/(4)),
circle(3,-2,0.05),
circle(3,-2,0.08),
circle(3,-2,0.10)
)}}} Graph of {{{-x+4y=-11}}} (red) and {{{2x+4y=-2}}} (green)