Question 147695

Start with the given system of equations:

{{{system(15x+3y=-12,2x+y=8)}}}



{{{-3(2x+y)=-3(8)}}} Multiply the both sides of the second equation by -3.



{{{-6x-3y=-24}}} Distribute and multiply.



So we have the new system of equations:

{{{system(15x+3y=-12,-6x-3y=-24)}}}



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+3y)+(-6x-3y)=(-12)+(-24)}}}



{{{(15x+-6x)+(3y+-3y)=-12+-24}}} Group like terms.



{{{9x+0y=-36}}} Combine like terms. Notice how the y terms cancel out.



{{{9x=-36}}} Simplify.



{{{x=(-36)/(9)}}} Divide both sides by {{{9}}} to isolate {{{x}}}.



{{{x=-4}}} Reduce.



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{{{15x+3y=-12}}} Now go back to the first equation.



{{{15(-4)+3y=-12}}} Plug in {{{x=-4}}}.



{{{-60+3y=-12}}} Multiply.



{{{3y=-12+60}}} Add {{{60}}} to both sides.



{{{3y=48}}} Combine like terms on the right side.



{{{y=(48)/(3)}}} Divide both sides by {{{3}}} to isolate {{{y}}}.



{{{y=16}}} Reduce.



So our answer is {{{x=-4}}} and {{{y=16}}}.



Which form the ordered pair *[Tex \LARGE \left(-4,16\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(-4,16\right)]. So this visually verifies our answer.



{{{drawing(500,500,-14,6,-5,18,
grid(1),
graph(500,500,-14,6,-5,18,(-12-15x)/(3),8-2x),
circle(-4,16,0.05),
circle(-4,16,0.08),
circle(-4,16,0.10)
)}}} Graph of {{{15x+3y=-12}}} (red) and {{{2x+y=8}}} (green) 



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Check:




{{{15x+3y=-12}}} Start with the first equation.



{{{15*(-4)+3*(16)=-12}}} Plug in {{{x=-4}}} and {{{y=16}}}.



{{{-12=-12}}} Evaluate and simplify the left side.



Since the equation is <b>true</b>, this means that (-4,16) is a solution of the first equation



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{{{2x+y=8}}} Start with the second equation.



{{{2*(-4)+(16)=8}}} Plug in {{{x=-4}}} and {{{y=16}}}.



{{{8=8}}} Evaluate and simplify the left side.



Since the equation is <b>true</b>, this means that (-4,16) is a solution of the second equation.



Since <font size="4"><b>all</b></font> of the equations of the system are true, this means that (-4,16) is a solution to the system. So this verifies our answer.