Question 221568

Start with the given system of equations:



{{{system(4x+2y=-6,2x+3y=-3)}}}



{{{4x+2y=-6}}} Start with the first equation.



{{{2y=-6-4x}}} Subtract {{{4x}}} from both sides.



{{{y=(-6-4x)/(2)}}} Divide both sides by {{{2}}} to isolate {{{y}}}.



{{{y=-2x-3}}} Rearrange the terms and simplify.



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



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



{{{2x-6x-9=-3}}} Distribute.



{{{-4x-9=-3}}} Combine like terms on the left side.



{{{-4x=-3+9}}} Add {{{9}}} to both sides.



{{{-4x=6}}} Combine like terms on the right side.



{{{x=(6)/(-4)}}} Divide both sides by {{{-4}}} to isolate {{{x}}}.



{{{x=-3/2}}} Reduce.



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Since we know that {{{x=-3/2}}}, we can use this to find {{{y}}}.



{{{4x+2y=-6}}} Go back to the first equation.



{{{4(-3/2)+2y=-6}}} Plug in {{{x=-3/2}}}.



{{{-6+2y=-6}}} Multiply.



{{{2y=-6+6}}} Add {{{6}}} to both sides.



{{{2y=0}}} Combine like terms on the right side.



{{{y=(0)/(2)}}} Divide both sides by {{{2}}} to isolate {{{y}}}.



{{{y=0}}} Reduce.



So the solutions are {{{x=-3/2}}} and {{{y=0}}}.



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(\frac{-3}{2},0\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11.5,8.5,-10,10,
grid(1),
graph(500,500,-11.5,8.5,-10,10,(-6-4x)/(2),(-3-2x)/(3)),
circle(-3/2,0,0.05),
circle(-3/2,0,0.08),
circle(-3/2,0,0.10)
)}}} Graph of {{{4x+2y=-6}}} (red) and {{{2x+3y=-3}}} (green)