Question 166281


Start with the given system of equations:

{{{system(-x-5y=-1,2x+3y=-5)}}}



{{{2(-x-5y)=2(-1)}}} Multiply the both sides of the first equation by 2.



{{{-2x-10y=-2}}} Distribute and multiply.



So we have the new system of equations:

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



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-10y)+(2x+3y)=(-2)+(-5)}}}



{{{(-2x+2x)+(-10y+3y)=-2+-5}}} Group like terms.



{{{0x+-7y=-7}}} Combine like terms. Notice how the x terms cancel out.



{{{-7y=-7}}} Simplify.



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



{{{y=1}}} Reduce.



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{{{-2x-10y=-2}}} Now go back to the first equation.



{{{-2x-10(1)=-2}}} Plug in {{{y=1}}}.



{{{-2x-10=-2}}} Multiply.



{{{-2x=-2+10}}} Add {{{10}}} to both sides.



{{{-2x=8}}} Combine like terms on the right side.



{{{x=(8)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{x}}}.



{{{x=-4}}} Reduce.



So our answer is {{{x=-4}}} and {{{y=1}}}.



Which form the ordered pair *[Tex \LARGE \left(-4,1\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(-4,1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-14,6,-9,11,
grid(1),
graph(500,500,-14,6,-9,11,(-1+x)/(-5),(-5-2x)/(3)),
circle(-4,1,0.05),
circle(-4,1,0.08),
circle(-4,1,0.10)
)}}} Graph of {{{-x-5y=-1}}} (red) and {{{2x+3y=-5}}} (green)