Question 182609

{{{system(2x-3y=9,-5x-3y=30)}}}

{{{-1(-5x-3y)=-1(30)}}} Multiply the both sides of the second equation by -1.

{{{5x+3y=-30}}} Distribute and multiply.

So we have the new system of equations:

{{{system(2x-3y=9,5x+3y=-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:

{{{(2x-3y)+(5x+3y)=(9)+(-30)}}}

{{{(2x+5x)+(-3y+3y)=9+-30}}} Group like terms.

{{{7x+0y=-21}}} Combine like terms.

{{{7x=-21}}} Simplify.

{{{x=(-21)/(7)}}} Divide both sides by {{{7}}} to isolate {{{x}}}.

{{{x=-3}}} Reduce.

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{{{2x-3y=9}}} Now go back to the first equation.

{{{2(-3)-3y=9}}} Plug in {{{x=-3}}}.

{{{-6-3y=9}}} Multiply.

{{{-3y=9+6}}} Add {{{6}}} to both sides.

{{{-3y=15}}} Combine like terms on the right side.

{{{y=(15)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{y}}}.

{{{y=-5}}} Reduce.

So our answer is {{{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,-13,7,-15,5,
grid(1),
graph(500,500,-13,7,-15,5,(9-2x)/(-3),(30+5x)/(-3)),
circle(-3,-5,0.05),
circle(-3,-5,0.08),
circle(-3,-5,0.10)
)}}} Graph of {{{2x-3y=9}}} (red) and {{{-5x-3y=30}}} (green)