Question 225185
# 1




Start with the given system of equations:

{{{system(10x+6y=0,-7x+2y=31)}}}



{{{-3(-7x+2y)=-3(31)}}} Multiply the both sides of the second equation by -3.



{{{21x-6y=-93}}} Distribute and multiply.



So we have the new system of equations:

{{{system(10x+6y=0,21x-6y=-93)}}}



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+6y)+(21x-6y)=(0)+(-93)}}}



{{{(10x+21x)+(6y+-6y)=0+-93}}} Group like terms.



{{{31x+0y=-93}}} Combine like terms.



{{{31x=-93}}} Simplify.



{{{x=(-93)/(31)}}} Divide both sides by {{{31}}} to isolate {{{x}}}.



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



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



{{{10x+6y=0}}} Now go back to the first equation.



{{{10(-3)+6y=0}}} Plug in {{{x=-3}}}.



{{{-30+6y=0}}} Multiply.



{{{6y=0+30}}} Add {{{30}}} to both sides.



{{{6y=30}}} Combine like terms on the right side.



{{{y=(30)/(6)}}} Divide both sides by {{{6}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



So the solutions are {{{x=-3}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(-3,5\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(-3,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-13,7,-5,15,
grid(1),
graph(500,500,-13,7,-5,15,(0-10x)/(6),(31+7x)/(2)),
circle(-3,5,0.05),
circle(-3,5,0.08),
circle(-3,5,0.10)
)}}} Graph of {{{10x+6y=0}}} (red) and {{{-7x+2y=31}}} (green) 



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# 2





Start with the given system of equations:

{{{system(3x+4y=-13,5x+6y=-19)}}}



{{{3(3x+4y)=3(-13)}}} Multiply the both sides of the first equation by 3.



{{{9x+12y=-39}}} Distribute and multiply.



{{{-2(5x+6y)=-2(-19)}}} Multiply the both sides of the second equation by -2.



{{{-10x-12y=38}}} Distribute and multiply.



So we have the new system of equations:

{{{system(9x+12y=-39,-10x-12y=38)}}}



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



{{{(9x+12y)+(-10x-12y)=(-39)+(38)}}}



{{{(9x+-10x)+(12y+-12y)=-39+38}}} Group like terms.



{{{-x+0y=-1}}} Combine like terms.



{{{-x=-1}}} Simplify.



{{{x=(-1)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{x}}}.



{{{x=1}}} Reduce.



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



{{{9x+12y=-39}}} Now go back to the first equation.



{{{9(1)+12y=-39}}} Plug in {{{x=1}}}.



{{{9+12y=-39}}} Multiply.



{{{12y=-39-9}}} Subtract {{{9}}} from both sides.



{{{12y=-48}}} Combine like terms on the right side.



{{{y=(-48)/(12)}}} Divide both sides by {{{12}}} to isolate {{{y}}}.



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



So the solutions are {{{x=1}}} and {{{y=-4}}}.



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



{{{drawing(500,500,-9,11,-14,6,
grid(1),
graph(500,500,-9,11,-14,6,(-13-3x)/(4),(-19-5x)/(6)),
circle(1,-4,0.05),
circle(1,-4,0.08),
circle(1,-4,0.10)
)}}} Graph of {{{3x+4y=-13}}} (red) and {{{5x+6y=-19}}} (green)