Question 550713


Start with the given system of equations:

{{{system(-8x-4y=-68,x+3y=16)}}}



{{{8(x+3y)=8(16)}}} Multiply the both sides of the second equation by 8.



{{{8x+24y=128}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-8x-4y=-68,8x+24y=128)}}}



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



{{{(-8x-4y)+(8x+24y)=(-68)+(128)}}}



{{{(-8x+8x)+(-4y+24y)=-68+128}}} Group like terms.



{{{0x+20y=60}}} Combine like terms.



{{{20y=60}}} Simplify.



{{{y=(60)/(20)}}} Divide both sides by {{{20}}} to isolate {{{y}}}.



{{{y=3}}} Reduce.



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{{{-8x-4y=-68}}} Now go back to the first equation.



{{{-8x-4(3)=-68}}} Plug in {{{y=3}}}.



{{{-8x-12=-68}}} Multiply.



{{{-8x=-68+12}}} Add {{{12}}} to both sides.



{{{-8x=-56}}} Combine like terms on the right side.



{{{x=(-56)/(-8)}}} Divide both sides by {{{-8}}} to isolate {{{x}}}.



{{{x=7}}} Reduce.



So the solutions are {{{x=7}}} and {{{y=3}}}.



Which form the ordered pair *[Tex \LARGE \left(7,3\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(7,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-3,17,-7,13,
grid(1),
graph(500,500,-3,17,-7,13,(-68+8x)/(-4),(16-x)/(3)),
circle(7,3,0.05),
circle(7,3,0.08),
circle(7,3,0.10)
)}}} Graph of {{{-8x-4y=-68}}} (red) and {{{x+3y=16}}} (green)