Question 264568


Start with the given system of equations:

{{{system(-4x+9y=9,x-3y=-6)}}}



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



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



So we have the new system of equations:

{{{system(-4x+9y=9,3x-9y=-18)}}}



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



{{{(-4x+9y)+(3x-9y)=(9)+(-18)}}}



{{{(-4x+3x)+(9y+-9y)=9+-18}}} Group like terms.



{{{-x+0y=-9}}} Combine like terms.



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



{{{x=(-9)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{x}}}.



{{{x=9}}} Reduce.



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{{{-4x+9y=9}}} Now go back to the first equation.



{{{-4(9)+9y=9}}} Plug in {{{x=9}}}.



{{{-36+9y=9}}} Multiply.



{{{9y=9+36}}} Add {{{36}}} to both sides.



{{{9y=45}}} Combine like terms on the right side.



{{{y=(45)/(9)}}} Divide both sides by {{{9}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



So the solutions are {{{x=9}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(9,5\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(9,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-1,19,-5,15,
grid(1),
graph(500,500,-1,19,-5,15,(9+4x)/(9),(-6-x)/(-3)),
circle(9,5,0.05),
circle(9,5,0.08),
circle(9,5,0.10)
)}}} Graph of {{{-4x+9y=9}}} (red) and {{{x-3y=-6}}} (green)