Question 678365


Start with the given system of equations:

{{{system(-7x+6y=-11,-5x-4y=17)}}}



{{{2(-7x+6y)=2(-11)}}} Multiply the both sides of the first equation by 2.



{{{-14x+12y=-22}}} Distribute and multiply.



{{{3(-5x-4y)=3(17)}}} Multiply the both sides of the second equation by 3.



{{{-15x-12y=51}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-14x+12y=-22,-15x-12y=51)}}}



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



{{{(-14x+12y)+(-15x-12y)=(-22)+(51)}}}



{{{(-14x+-15x)+(12y+-12y)=-22+51}}} Group like terms.



{{{-29x+0y=29}}} Combine like terms.



{{{-29x=29}}} Simplify.



{{{x=(29)/(-29)}}} Divide both sides by {{{-29}}} to isolate {{{x}}}.



{{{x=-1}}} Reduce.



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{{{-14x+12y=-22}}} Now go back to the first equation.



{{{-14(-1)+12y=-22}}} Plug in {{{x=-1}}}.



{{{14+12y=-22}}} Multiply.



{{{12y=-22-14}}} Subtract {{{14}}} from both sides.



{{{12y=-36}}} Combine like terms on the right side.



{{{y=(-36)/(12)}}} Divide both sides by {{{12}}} to isolate {{{y}}}.



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



So the solutions are {{{x=-1}}} and {{{y=-3}}}.



Which form the ordered pair *[Tex \LARGE \left(-1,-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(-1,-3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11,9,-13,7,
grid(1),
graph(500,500,-11,9,-13,7,(-11+7x)/(6),(17+5x)/(-4)),
circle(-1,-3,0.05),
circle(-1,-3,0.08),
circle(-1,-3,0.10)
)}}} Graph of {{{-7x+6y=-11}}} (red) and {{{-5x-4y=17}}} (green)