Question 299353


Start with the given system of equations:

{{{system(-4x-3y=3,-2x+5y=21)}}}



{{{-2(-2x+5y)=-2(21)}}} Multiply the both sides of the second equation by -2.



{{{4x-10y=-42}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-4x-3y=3,4x-10y=-42)}}}



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



{{{(-4x-3y)+(4x-10y)=(3)+(-42)}}}



{{{(-4x+4x)+(-3y+-10y)=3+-42}}} Group like terms.



{{{0x+-13y=-39}}} Combine like terms.



{{{-13y=-39}}} Simplify.



{{{y=(-39)/(-13)}}} Divide both sides by {{{-13}}} to isolate {{{y}}}.



{{{y=3}}} Reduce.



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{{{-4x-3y=3}}} Now go back to the first equation.



{{{-4x-3(3)=3}}} Plug in {{{y=3}}}.



{{{-4x-9=3}}} Multiply.



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



{{{-4x=12}}} Combine like terms on the right side.



{{{x=(12)/(-4)}}} Divide both sides by {{{-4}}} to isolate {{{x}}}.



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



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



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



{{{drawing(500,500,-13,7,-7,13,
grid(1),
graph(500,500,-13,7,-7,13,(3+4x)/(-3),(21+2x)/(5)),
circle(-3,3,0.05),
circle(-3,3,0.08),
circle(-3,3,0.10)
)}}} Graph of {{{-4x-3y=3}}} (red) and {{{-2x+5y=21}}} (green)