Question 189588
Start with the given system of equations:

{{{system(x+2y=10,3x+4y=8)}}}



{{{-3(x+2y)=-3(10)}}} Multiply the both sides of the first equation by -3.



{{{-3x-6y=-30}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-3x-6y=-30,3x+4y=8)}}}



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-6y)+(3x+4y)=(-30)+(8)}}}



{{{(-3x+3x)+(-6y+4y)=-30+8}}} Group like terms.



{{{0x+-2y=-22}}} Combine like terms.



{{{-2y=-22}}} Simplify.



{{{y=(-22)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{y}}}.



{{{y=11}}} Reduce.



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{{{-3x-6y=-30}}} Now go back to the first equation.



{{{-3x-6(11)=-30}}} Plug in {{{y=11}}}.



{{{-3x-66=-30}}} Multiply.



{{{-3x=-30+66}}} Add {{{66}}} to both sides.



{{{-3x=36}}} Combine like terms on the right side.



{{{x=(36)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{x}}}.



{{{x=-12}}} Reduce.



So the solutions are {{{x=-12}}} and {{{y=11}}}.



Which form the ordered pair *[Tex \LARGE \left(-12,11\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(-12,11\right)]. So this visually verifies our answer.



{{{drawing(500,500,-15,5,-5,15,
grid(1),
graph(500,500,-15,5,-5,15,(10-x)/(2),(8-3x)/(4)),
circle(-12,11,0.05),
circle(-12,11,0.08),
circle(-12,11,0.10)
)}}} Graph of {{{x+2y=10}}} (red) and {{{3x+4y=8}}} (green)