Question 244750

Start with the given system of equations:

{{{system(3x-5y=-3,-9x-15y=9)}}}



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



{{{9x-15y=-9}}} Distribute and multiply.



So we have the new system of equations:

{{{system(9x-15y=-9,-9x-15y=9)}}}



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



{{{(9x-15y)+(-9x-15y)=(-9)+(9)}}}



{{{(9x+-9x)+(-15y+-15y)=-9+9}}} Group like terms.



{{{0x+-30y=0}}} Combine like terms.



{{{-30y=0}}} Simplify.



{{{y=(0)/(-30)}}} Divide both sides by {{{-30}}} to isolate {{{y}}}.



{{{y=0}}} Reduce.



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{{{9x-15y=-9}}} Now go back to the first equation.



{{{9x-15(0)=-9}}} Plug in {{{y=0}}}.



{{{9x=-9}}} Multiply.



{{{x=(-9)/(9)}}} Divide both sides by {{{9}}} to isolate {{{x}}}.



{{{x=-1}}} Reduce.



So the solutions are {{{x=-1}}} and {{{y=0}}}.



Which form the ordered pair *[Tex \LARGE \left(-1,0\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(-1,0\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11,9,-10,10,
grid(1),
graph(500,500,-11,9,-10,10,(-3-3x)/(-5),(9+9x)/(-15)),
circle(-1,0,0.05),
circle(-1,0,0.08),
circle(-1,0,0.10)
)}}} Graph of {{{3x-5y=-3}}} (red) and {{{-9x-15y=9}}} (green)