Question 198012

Start with the given system of equations:

{{{system(12x-7y=-10,-4x+3y=10)}}}



{{{3(-4x+3y)=3(10)}}} Multiply the both sides of the second equation by 3.



{{{-12x+9y=30}}} Distribute and multiply.



So we have the new system of equations:

{{{system(12x-7y=-10,-12x+9y=30)}}}



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



{{{(12x-7y)+(-12x+9y)=(-10)+(30)}}}



{{{(12x+-12x)+(-7y+9y)=-10+30}}} Group like terms.



{{{0x+2y=20}}} Combine like terms.



{{{2y=20}}} Simplify.



{{{y=(20)/(2)}}} Divide both sides by {{{2}}} to isolate {{{y}}}.



{{{y=10}}} Reduce.



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{{{12x-7y=-10}}} Now go back to the first equation.



{{{12x-7(10)=-10}}} Plug in {{{y=10}}}.



{{{12x-70=-10}}} Multiply.



{{{12x=-10+70}}} Add {{{70}}} to both sides.



{{{12x=60}}} Combine like terms on the right side.



{{{x=(60)/(12)}}} Divide both sides by {{{12}}} to isolate {{{x}}}.



{{{x=5}}} Reduce.



So the solutions are {{{x=5}}} and {{{y=10}}}.



Which form the ordered pair *[Tex \LARGE \left(5,10\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(5,10\right)]. So this visually verifies our answer.



{{{drawing(500,500,-5,15,-5,15,
grid(1),
graph(500,500,-5,15,-5,15,(-10-12x)/(-7),(10+4x)/(3)),
circle(5,10,0.05),
circle(5,10,0.08),
circle(5,10,0.10)
)}}} Graph of {{{12x-7y=-10}}} (red) and {{{-4x+3y=10}}} (green)