Question 264538
I'll do the first one to get you started.





Start with the given system of equations:

{{{system(3x-2y=12,2x+3y=-5)}}}



{{{3(3x-2y)=3(12)}}} Multiply the both sides of the first equation by 3.



{{{9x-6y=36}}} Distribute and multiply.



{{{2(2x+3y)=2(-5)}}} Multiply the both sides of the second equation by 2.



{{{4x+6y=-10}}} Distribute and multiply.



So we have the new system of equations:

{{{system(9x-6y=36,4x+6y=-10)}}}



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



{{{(9x-6y)+(4x+6y)=(36)+(-10)}}}



{{{(9x+4x)+(-6y+6y)=36+-10}}} Group like terms.



{{{13x+0y=26}}} Combine like terms.



{{{13x=26}}} Simplify.



{{{x=(26)/(13)}}} Divide both sides by {{{13}}} to isolate {{{x}}}.



{{{x=2}}} Reduce.



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{{{9x-6y=36}}} Now go back to the first equation.



{{{9(2)-6y=36}}} Plug in {{{x=2}}}.



{{{18-6y=36}}} Multiply.



{{{-6y=36-18}}} Subtract {{{18}}} from both sides.



{{{-6y=18}}} Combine like terms on the right side.



{{{y=(18)/(-6)}}} Divide both sides by {{{-6}}} to isolate {{{y}}}.



{{{y=-3}}} Reduce.



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



Which form the ordered pair *[Tex \LARGE \left(2,-3\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(2,-3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-8,12,-13,7,
grid(1),
graph(500,500,-8,12,-13,7,(12-3x)/(-2),(-5-2x)/(3)),
circle(2,-3,0.05),
circle(2,-3,0.08),
circle(2,-3,0.10)
)}}} Graph of {{{3x-2y=12}}} (red) and {{{2x+3y=-5}}} (green)