Question 178005


Start with the given system of equations:

{{{system(x+6y=-4,3x-4y=10)}}}



{{{-3(x+6y)=-3(-4)}}} Multiply the both sides of the first equation by -3.



{{{-3x-18y=12}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-3x-18y=12,3x-4y=10)}}}



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



{{{(-3x-18y)+(3x-4y)=(12)+(10)}}}



{{{(-3x+3x)+(-18y+-4y)=12+10}}} Group like terms.



{{{0x+-22y=22}}} Combine like terms.



{{{-22y=22}}} Simplify.



{{{y=(22)/(-22)}}} Divide both sides by {{{-22}}} to isolate {{{y}}}.



{{{y=-1}}} Reduce.



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{{{-3x-18y=12}}} Now go back to the first equation.



{{{-3x-18(-1)=12}}} Plug in {{{y=-1}}}.



{{{-3x+18=12}}} Multiply.



{{{-3x=12-18}}} Subtract {{{18}}} from both sides.



{{{-3x=-6}}} Combine like terms on the right side.



{{{x=(-6)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{x}}}.



{{{x=2}}} Reduce.



So our answer is {{{x=2}}} and {{{y=-1}}}.



Which form the ordered pair *[Tex \LARGE \left(2,-1\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,-1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-8,12,-11,9,
grid(1),
graph(500,500,-8,12,-11,9,(-4-x)/(6),(10-3x)/(-4)),
circle(2,-1,0.05),
circle(2,-1,0.08),
circle(2,-1,0.10)
)}}} Graph of {{{x+6y=-4}}} (red) and {{{3x-4y=10}}} (green)