Question 177480

Start with the given system of equations:

{{{system(2x-3y=-3,x+6y=-9)}}}



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



{{{4x-6y=-6}}} Distribute and multiply.



So we have the new system of equations:

{{{system(4x-6y=-6,x+6y=-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:



{{{(4x-6y)+(x+6y)=(-6)+(-9)}}}



{{{(4x+1x)+(-6y+6y)=-6+-9}}} Group like terms.



{{{5x+0y=-15}}} Combine like terms.



{{{5x=-15}}} Simplify.



{{{x=(-15)/(5)}}} Divide both sides by {{{5}}} to isolate {{{x}}}.



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



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{{{4x-6y=-6}}} Now go back to the first equation.



{{{4(-3)-6y=-6}}} Plug in {{{x=-3}}}.



{{{-12-6y=-6}}} Multiply.



{{{-6y=-6+12}}} Add {{{12}}} to both sides.



{{{-6y=6}}} Combine like terms on the right side.



{{{y=(6)/(-6)}}} Divide both sides by {{{-6}}} to isolate {{{y}}}.



{{{y=-1}}} Reduce.



So our answer is {{{x=-3}}} and {{{y=-1}}}.



Which form the ordered pair *[Tex \LARGE \left(-3,-1\right)]. This is the point that makes both equations true.



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,-1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-13,7,-11,9,
grid(1),
graph(500,500,-13,7,-11,9,(-3-2x)/(-3),(-9-x)/(6)),
circle(-3,-1,0.05),
circle(-3,-1,0.08),
circle(-3,-1,0.10)
)}}} Graph of {{{2x-3y=-3}}} (red) and {{{x+6y=-9}}} (green)