Question 182609


Start with the given system of equations:

{{{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)