Question 299401


Start with the given system of equations:

{{{system(5x+7y=-1,7x+3y=-15)}}}



{{{3(5x+7y)=3(-1)}}} Multiply the both sides of the first equation by 3.



{{{15x+21y=-3}}} Distribute and multiply.



{{{-7(7x+3y)=-7(-15)}}} Multiply the both sides of the second equation by -7.



{{{-49x-21y=105}}} Distribute and multiply.



So we have the new system of equations:

{{{system(15x+21y=-3,-49x-21y=105)}}}



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



{{{(15x+21y)+(-49x-21y)=(-3)+(105)}}}



{{{(15x+-49x)+(21y+-21y)=-3+105}}} Group like terms.



{{{-34x+0y=102}}} Combine like terms.



{{{-34x=102}}} Simplify.



{{{x=(102)/(-34)}}} Divide both sides by {{{-34}}} to isolate {{{x}}}.



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



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{{{15x+21y=-3}}} Now go back to the first equation.



{{{15(-3)+21y=-3}}} Plug in {{{x=-3}}}.



{{{-45+21y=-3}}} Multiply.



{{{21y=-3+45}}} Add {{{45}}} to both sides.



{{{21y=42}}} Combine like terms on the right side.



{{{y=(42)/(21)}}} Divide both sides by {{{21}}} to isolate {{{y}}}.



{{{y=2}}} Reduce.



So the solutions are {{{x=-3}}} and {{{y=2}}}.



Which form the ordered pair *[Tex \LARGE \left(-3,2\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(-3,2\right)]. So this visually verifies our answer.



{{{drawing(500,500,-13,7,-8,12,
grid(1),
graph(500,500,-13,7,-8,12,(-1-5x)/(7),(-15-7x)/(3)),
circle(-3,2,0.05),
circle(-3,2,0.08),
circle(-3,2,0.10)
)}}} Graph of {{{5x+7y=-1}}} (red) and {{{7x+3y=-15}}} (green)