Question 236841

Start with the given system of equations:

{{{system(-x+2y=16,x-3y=-21)}}}



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



{{{(-1x+2y)+(x-3y)=(16)+(-21)}}}



{{{(-1x+1x)+(2y+-3y)=16+-21}}} Group like terms.



{{{0x+-y=-5}}} Combine like terms. Notice how the x terms cancel out.



{{{-y=-5}}} Simplify.



{{{y=(-5)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



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{{{-1x+2y=16}}} Now go back to the first equation.



{{{-1x+2(5)=16}}} Plug in {{{y=5}}}.



{{{-1x+10=16}}} Multiply.



{{{-x=16-10}}} Subtract {{{10}}} from both sides.



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



{{{x=(6)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{x}}}.



{{{x=-6}}} Reduce.



So our answer is {{{x=-6}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(-6,5\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(-6,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-16,4,-5,15,
grid(1),
graph(500,500,-16,4,-5,15,(16+x)/(2),(-21-x)/(-3)),
circle(-6,5,0.05),
circle(-6,5,0.08),
circle(-6,5,0.10)
)}}} Graph of {{{-x+2y=16}}} (red) and {{{x-3y=-21}}} (green)