Question 202454


Start with the given system of equations:

{{{system(x-4y=29,x-y=5)}}}



{{{-1(x-y)=-1(5)}}} Multiply the both sides of the second equation by -1.



{{{-x+y=-5}}} Distribute and multiply.



So we have the new system of equations:

{{{system(x-4y=29,-x+y=-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:



{{{(x-4y)+(-x+y)=(29)+(-5)}}}



{{{(x-x)+(-4y+y)=29+-5}}} Group like terms.



{{{0x-3y=24}}} Combine like terms.



{{{-3y=24}}} Simplify.



{{{y=(24)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{y}}}.



{{{y=-8}}} Reduce.



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{{{x-4y=29}}} Now go back to the first equation.



{{{x-4(-8)=29}}} Plug in {{{y=-8}}}.



{{{x+32=29}}} Multiply.



{{{x=29-32}}} Subtract {{{32}}} from both sides.



{{{x=-3}}} Combine like terms on the right side.



So the solutions are {{{x=-3}}} and {{{y=-8}}}.



Which form the ordered pair *[Tex \LARGE \left(-3,-8\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,-8\right)]. So this visually verifies our answer.



{{{drawing(500,500,-13,7,-18,2,
grid(1),
graph(500,500,-13,7,-18,2,(29-x)/(-4),(5-x)/(-1)),
circle(-3,-8,0.05),
circle(-3,-8,0.08),
circle(-3,-8,0.10)
)}}} Graph of {{{x-4y=29}}} (red) and {{{x-y=5}}} (green)