Question 175938


Start with the given system of equations:

{{{system(x-y=-1,2x-3y=-5)}}}



{{{-3(x-y)=-3(-1)}}} Multiply the both sides of the first equation by -3.



{{{-3x+3y=3}}} Distribute and multiply.



So we have the new system of equations:

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



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



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



{{{(-3x+2x)+(3y+-3y)=3+-5}}} Group like terms.



{{{-x+0y=-2}}} Combine like terms.



{{{-x=-2}}} Simplify.



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



{{{x=2}}} Reduce.



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



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



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



{{{3y=3+6}}} Add {{{6}}} to both sides.



{{{3y=9}}} Combine like terms on the right side.



{{{y=(9)/(3)}}} Divide both sides by {{{3}}} to isolate {{{y}}}.



{{{y=3}}} Reduce.



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