Question 281923

Start with the given system of equations:

{{{system(x+5y=-6,2x+5y=3)}}}



{{{-1(x+5y)=-1(-6)}}} Multiply the both sides of the first equation by -1.



{{{-x-5y=6}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-x-5y=6,2x+5y=3)}}}



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



{{{(-x-5y)+(2x+5y)=(6)+(3)}}}



{{{(-x+2x)+(-5y+5y)=6+3}}} Group like terms.



{{{x+0y=9}}} Combine like terms.



{{{x=9}}} Simplify.



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{{{-x-5y=6}}} Now go back to the first equation.



{{{-9-5y=6}}} Plug in {{{x=9}}}.



{{{-9-5y=6}}} Multiply.



{{{-5y=6+9}}} Add {{{9}}} to both sides.



{{{-5y=15}}} Combine like terms on the right side.



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



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



So the solutions are {{{x=9}}} and {{{y=-3}}}.



Which form the ordered pair *[Tex \LARGE \left(9,-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(9,-3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-1,19,-13,7,
grid(1),
graph(500,500,-1,19,-13,7,(-6-x)/(5),(3-2x)/(5)),
circle(9,-3,0.05),
circle(9,-3,0.08),
circle(9,-3,0.10)
)}}} Graph of {{{x+5y=-6}}} (red) and {{{2x+5y=3}}} (green)