Question 376347


Start with the given system of equations:

{{{system(9x+2y=-51,-9x+y=69)}}}



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



{{{(9x+2y)+(-9x+y)=(-51)+(69)}}}



{{{(9x+-9x)+(2y+1y)=-51+69}}} Group like terms.



{{{0x+3y=18}}} Combine like terms.



{{{3y=18}}} Simplify.



{{{y=(18)/(3)}}} Divide both sides by {{{3}}} to isolate {{{y}}}.



{{{y=6}}} Reduce.



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{{{9x+2y=-51}}} Now go back to the first equation.



{{{9x+2(6)=-51}}} Plug in {{{y=6}}}.



{{{9x+12=-51}}} Multiply.



{{{9x=-51-12}}} Subtract {{{12}}} from both sides.



{{{9x=-63}}} Combine like terms on the right side.



{{{x=(-63)/(9)}}} Divide both sides by {{{9}}} to isolate {{{x}}}.



{{{x=-7}}} Reduce.



So the solutions are {{{x=-7}}} and {{{y=6}}}.



Which form the ordered pair *[Tex \LARGE \left(-7,6\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(-7,6\right)]. So this visually verifies our answer.



{{{drawing(500,500,-17,3,-4,16,
grid(1),
graph(500,500,-17,3,-4,16,(-51-9x)/(2),69+9x),
circle(-7,6,0.05),
circle(-7,6,0.08),
circle(-7,6,0.10)
)}}} Graph of {{{9x+2y=-51}}} (red) and {{{-9x+y=69}}} (green) 



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Jim