Question 347764

Start with the given system of equations:

{{{system(6x-y=3,5x+3y=-9)}}}



{{{3(6x-y)=3(3)}}} Multiply the both sides of the first equation by 3.



{{{18x-3y=9}}} Distribute and multiply.



So we have the new system of equations:

{{{system(18x-3y=9,5x+3y=-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:



{{{(18x-3y)+(5x+3y)=(9)+(-9)}}}



{{{(18x+5x)+(-3y+3y)=9+-9}}} Group like terms.



{{{23x+0y=0}}} Combine like terms.



{{{23x=0}}} Simplify.



{{{x=(0)/(23)}}} Divide both sides by {{{23}}} to isolate {{{x}}}.



{{{x=0}}} Reduce.



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{{{18x-3y=9}}} Now go back to the first equation.



{{{18(0)-3y=9}}} Plug in {{{x=0}}}.



{{{0-3y=9}}} Multiply.



{{{-3y=9-0}}} Subtract {{{0}}} from both sides.



{{{-3y=9}}} Combine like terms on the right side.



{{{y=(9)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{y}}}.



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



So the solutions are {{{x=0}}} and {{{y=-3}}}.



Which form the ordered pair *[Tex \LARGE \left(0,-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(0,-3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-10,10,-13,7,
grid(1),
graph(500,500,-10,10,-13,7,(3-6x)/(-1),(-9-5x)/(3)),
circle(0,-3,0.05),
circle(0,-3,0.08),
circle(0,-3,0.10)
)}}} Graph of {{{6x-y=3}}} (red) and {{{5x+3y=-9}}} (green) 



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