Question 200628


Start with the given system of equations:

{{{system(8x+9y=-2,-9x+y=69)}}}



{{{-9(-9x+y)=-9(69)}}} Multiply the both sides of the second equation by -9.



{{{81x-9y=-621}}} Distribute and multiply.



So we have the new system of equations:

{{{system(8x+9y=-2,81x-9y=-621)}}}



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



{{{(8x+9y)+(81x-9y)=(-2)+(-621)}}}



{{{(8x+81x)+(9y+-9y)=-2+-621}}} Group like terms.



{{{89x+0y=-623}}} Combine like terms.



{{{89x=-623}}} Simplify.



{{{x=(-623)/(89)}}} Divide both sides by {{{89}}} to isolate {{{x}}}.



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



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{{{8x+9y=-2}}} Now go back to the first equation.



{{{8(-7)+9y=-2}}} Plug in {{{x=-7}}}.



{{{-56+9y=-2}}} Multiply.



{{{9y=-2+56}}} Add {{{56}}} to both sides.



{{{9y=54}}} Combine like terms on the right side.



{{{y=(54)/(9)}}} Divide both sides by {{{9}}} to isolate {{{y}}}.



{{{y=6}}} 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,(-2-8x)/(9),69+9x),
circle(-7,6,0.05),
circle(-7,6,0.08),
circle(-7,6,0.10)
)}}} Graph of {{{8x+9y=-2}}} (red) and {{{-9x+y=69}}} (green)