Question 152165


Start with the given system of equations:

{{{system(2x+3y=-16,5x-10y=30)}}}



{{{5(2x+3y)=5(-16)}}} Multiply the both sides of the first equation by 5.



{{{10x+15y=-80}}} Distribute and multiply.



{{{-2(5x-10y)=-2(30)}}} Multiply the both sides of the second equation by -2.



{{{-10x+20y=-60}}} Distribute and multiply.



So we have the new system of equations:

{{{system(10x+15y=-80,-10x+20y=-60)}}}



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



{{{(10x+15y)+(-10x+20y)=(-80)+(-60)}}}



{{{(10x+-10x)+(15y+20y)=-80+-60}}} Group like terms.



{{{0x+35y=-140}}} Combine like terms. Notice how the x terms cancel out.



{{{35y=-140}}} Simplify.



{{{y=(-140)/(35)}}} Divide both sides by {{{35}}} to isolate {{{y}}}.



{{{y=-4}}} Reduce.



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{{{10x+15y=-80}}} Now go back to the first equation.



{{{10x+15(-4)=-80}}} Plug in {{{y=-4}}}.



{{{10x-60=-80}}} Multiply.



{{{10x=-80+60}}} Add {{{60}}} to both sides.



{{{10x=-20}}} Combine like terms on the right side.



{{{x=(-20)/(10)}}} Divide both sides by {{{10}}} to isolate {{{x}}}.



{{{x=-2}}} Reduce.



So our answer is {{{x=-2}}} and {{{y=-4}}}.



Which form the ordered pair *[Tex \LARGE \left(-2,-4\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(-2,-4\right)]. So this visually verifies our answer.



{{{drawing(500,500,-12,8,-14,6,
grid(1),
graph(500,500,-12,8,-14,6,(-16-2x)/(3),(30-5x)/(-10)),
circle(-2,-4,0.05),
circle(-2,-4,0.08),
circle(-2,-4,0.10)
)}}} Graph of {{{2x+3y=-16}}} (red) and {{{5x-10y=30}}} (green)