Question 254869


Start with the given system of equations:

{{{system(6x-2y=8,-4x+6y=18)}}}



{{{3(6x-2y)=3(8)}}} Multiply the both sides of the first equation by 3.



{{{18x-6y=24}}} Distribute and multiply.



So we have the new system of equations:

{{{system(18x-6y=24,-4x+6y=18)}}}



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



{{{(18x-6y)+(-4x+6y)=(24)+(18)}}}



{{{(18x+-4x)+(-6y+6y)=24+18}}} Group like terms.



{{{14x+0y=42}}} Combine like terms.



{{{14x=42}}} Simplify.



{{{x=(42)/(14)}}} Divide both sides by {{{14}}} to isolate {{{x}}}.



{{{x=3}}} Reduce.



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{{{18x-6y=24}}} Now go back to the first equation.



{{{18(3)-6y=24}}} Plug in {{{x=3}}}.



{{{54-6y=24}}} Multiply.



{{{-6y=24-54}}} Subtract {{{54}}} from both sides.



{{{-6y=-30}}} Combine like terms on the right side.



{{{y=(-30)/(-6)}}} Divide both sides by {{{-6}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



So the solutions are {{{x=3}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(3,5\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(3,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-7,13,-5,15,
grid(1),
graph(500,500,-7,13,-5,15,(8-6x)/(-2),(18+4x)/(6)),
circle(3,5,0.05),
circle(3,5,0.08),
circle(3,5,0.10)
)}}} Graph of {{{6x-2y=8}}} (red) and {{{-4x+6y=18}}} (green)