Question 203073


Start with the given system of equations:



{{{system(-3x+y=3,2x-y=-1)}}}



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



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



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



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{{{2x-y=-1}}} Move onto the second equation.



{{{2x-(3x+3)=-1}}} Now plug in {{{y=3x+3}}}.



{{{2x-3x-3=-1}}} Distribute.



{{{-x-3=-1}}} Combine like terms on the left side.



{{{-x=-1+3}}} Add {{{3}}} to both sides.



{{{-x=2}}} Combine like terms on the right side.



{{{x=(2)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{x}}}.



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



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Since we know that {{{x=-2}}}, we can use this to find {{{y}}}.



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



{{{-3(-2)+y=3}}} Plug in {{{x=-2}}}.



{{{6+y=3}}} Multiply.



{{{y=3-6}}} Subtract {{{6}}} from both sides.



{{{y=-3}}} Combine like terms on the right side.



So the solutions are {{{x=-2}}} and {{{y=-3}}}.



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,-3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-12,8,-13,7,
grid(1),
graph(500,500,-12,8,-13,7,3+3x,(-1-2x)/(-1)),
circle(-2,-3,0.05),
circle(-2,-3,0.08),
circle(-2,-3,0.10)
)}}} Graph of {{{-3x+y=3}}} (red) and {{{2x-y=-1}}} (green)