Question 148096


Start with the given system of equations:

{{{system(3x+4y=-15,3x+6y=-21)}}}



{{{-1(3x+6y)=-1(-21)}}} Multiply the both sides of the second equation by -1.



{{{-3x-6y=21}}} Distribute and multiply.



So we have the new system of equations:

{{{system(3x+4y=-15,-3x-6y=21)}}}



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



{{{(3x+4y)+(-3x-6y)=(-15)+(21)}}}



{{{(3x+-3x)+(4y+-6y)=-15+21}}} Group like terms.



{{{0x+-2y=6}}} Combine like terms. Notice how the x terms cancel out.



{{{-2y=6}}} Simplify.



{{{y=(6)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{y}}}.



{{{y=-3}}} Reduce.



------------------------------------------------------------------



{{{3x+4y=-15}}} Now go back to the first equation.



{{{3x+4(-3)=-15}}} Plug in {{{y=-3}}}.



{{{3x-12=-15}}} Multiply.



{{{3x=-15+12}}} Add {{{12}}} to both sides.



{{{3x=-3}}} Combine like terms on the right side.



{{{x=(-3)/(3)}}} Divide both sides by {{{3}}} to isolate {{{x}}}.



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



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



Which form the ordered pair *[Tex \LARGE \left(-1,-3\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,-3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11,9,-13,7,
grid(1),
graph(500,500,-11,9,-13,7,(-15-3x)/(4),(-21-3x)/(6)),
circle(-1,-3,0.05),
circle(-1,-3,0.08),
circle(-1,-3,0.10)
)}}} Graph of {{{3x+4y=-15}}} (red) and {{{3x+6y=-21}}} (green)