Question 207113


Start with the given system of equations:



{{{system(-3x+y=4,2x-3y=9)}}}



{{{-3x+y=4}}} Start with the first equation.



{{{y=4+3x}}} Add {{{3x}}} to both sides.



{{{y=3x+4}}} Rearrange the terms and simplify.



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{{{2x-3y=9}}} Move onto the second equation.



{{{2x-3(3x+4)=9}}} Now plug in {{{y=3x+4}}}.



{{{2x-9x-12=9}}} Distribute.



{{{-7x-12=9}}} Combine like terms on the left side.



{{{-7x=9+12}}} Add {{{12}}} to both sides.



{{{-7x=21}}} Combine like terms on the right side.



{{{x=(21)/(-7)}}} Divide both sides by {{{-7}}} to isolate {{{x}}}.



{{{x=-3}}} Reduce.



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Since we know that {{{x=-3}}}, we can use this to find {{{y}}}.



{{{-3x+y=4}}} Go back to the first equation.



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



{{{9+y=4}}} Multiply.



{{{y=4-9}}} Subtract {{{9}}} from both sides.



{{{y=-5}}} Combine like terms on the right side.



So the solutions are {{{x=-3}}} and {{{y=-5}}}.



which form the ordered pair (-3,-5)



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(-3,-5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-13,7,-15,5,
grid(1),
graph(500,500,-13,7,-15,5,4+3x,(9-2x)/(-3)),
circle(-3,-5,0.05),
circle(-3,-5,0.08),
circle(-3,-5,0.10)
)}}} Graph of {{{-3x+y=4}}} (red) and {{{2x-3y=9}}} (green)