Question 183398
I'm going to use elimination to solve this system:



{{{13x-3y=-50 }}} Start with the first equation



{{{5(13x-3y)=5(-50) }}} Multiply both sides by 5



{{{65x-15y=-250 }}} Distribute and multiply.


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{{{12x+5y=16}}} Move onto the second equation



{{{3(12x+5y)=3(16)}}} Multiply both sides by 3



{{{36x+15y=48}}} Distribute and multiply.



So we have the new system of equations:

{{{system(65x-15y=-250,36x+15y=48)}}}



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



{{{(65x-15y)+(36x+15y)=(-250)+(48)}}}



{{{(65x+36x)+(-15y+15y)=-250+48}}} Group like terms.



{{{101x+0y=-202}}} Combine like terms. Notice how the y terms cancel out.



{{{101x=-202}}} Simplify.



{{{x=(-202)/(101)}}} Divide both sides by {{{101}}} to isolate {{{x}}}.



{{{x=-2}}} Reduce.



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{{{65x-15y=-250}}} Now go back to the first equation.



{{{65(-2)-15y=-250}}} Plug in {{{x=-2}}}.



{{{-130-15y=-250}}} Multiply.



{{{-15y=-250+130}}} Add {{{130}}} to both sides.



{{{-15y=-120}}} Combine like terms on the right side.



{{{y=(-120)/(-15)}}} Divide both sides by {{{-15}}} to isolate {{{y}}}.



{{{y=8}}} Reduce.



So our answer is {{{x=-2}}} and {{{y=8}}}.



Which form the ordered pair *[Tex \LARGE \left(-2,8\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(-2,8\right)]. So this visually verifies our answer.



{{{drawing(500,500,-12,8,-2,18,
grid(1),
graph(500,500,-12,8,-2,18,(-50-13x)/(-3),(16-12x)/(5)),
circle(-2,8,0.05),
circle(-2,8,0.08),
circle(-2,8,0.10)
)}}} Graph of {{{13x-3y=-50}}} (red) and {{{12x+5y=16}}} (green)