Question 173188


Start with the given system of equations:

{{{system(-4x+y=-5,2x+y=7)}}}



{{{2(2x+y)=2(7)}}} Multiply the both sides of the second equation by 2.



{{{4x+2y=14}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-4x+y=-5,4x+2y=14)}}}



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+y)+(4x+2y)=(-5)+(14)}}}



{{{(-4x+4x)+(1y+2y)=-5+14}}} Group like terms.



{{{0x+3y=9}}} Combine like terms. Notice how the x terms cancel out.



{{{3y=9}}} Simplify.



{{{y=(9)/(3)}}} Divide both sides by {{{3}}} to isolate {{{y}}}.



{{{y=3}}} Reduce.



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{{{-4x+y=-5}}} Now go back to the first equation.



{{{-4x+3=-5}}} Plug in {{{y=3}}}.



{{{-4x+3=-5}}} Multiply.



{{{-4x=-5-3}}} Subtract {{{3}}} from both sides.



{{{-4x=-8}}} Combine like terms on the right side.



{{{x=(-8)/(-4)}}} Divide both sides by {{{-4}}} to isolate {{{x}}}.



{{{x=2}}} Reduce.



So our answer is {{{x=2}}} and {{{y=3}}}.



Which form the ordered pair *[Tex \LARGE \left(2,3\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(2,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-8,12,-7,13,
grid(1),
graph(500,500,-8,12,-7,13,-5+4x,7-2x),
circle(2,3,0.05),
circle(2,3,0.08),
circle(2,3,0.10)
)}}} Graph of {{{-4x+y=-5}}} (red) and {{{2x+y=7}}} (green)