Question 182608

Start with the given system of equations:

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



{{{-1(x+4y)=-1(11)}}} Multiply the both sides of the first equation by -1.



{{{-x-4y=-11}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-x-4y=-11,x-6y=11)}}}



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-6y)=(-11)+(11)}}}



{{{(-x+x)+(-4y+-6y)=-11+11}}} Group like terms.



{{{0x+-10y=0}}} Combine like terms.



{{{-10y=0}}} Simplify.



{{{y=(0)/(-10)}}} Divide both sides by {{{-10}}} to isolate {{{y}}}.



{{{y=0}}} Reduce.



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{{{-x-4y=-11}}} Now go back to the first equation.



{{{-x-4(0)=-11}}} Plug in {{{y=0}}}.



{{{-x+0=-11}}} Multiply.



{{{-x=-11}}} Remove any zero terms.



{{{x=(-11)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{x}}}.



{{{x=11}}} Reduce.



So our answer is {{{x=11}}} and {{{y=0}}}.



Which form the ordered pair *[Tex \LARGE \left(11,0\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(11,0\right)]. So this visually verifies our answer.



{{{drawing(500,500,-5,15,-10,10,
grid(1),
graph(500,500,-5,15,-10,10,(11-x)/(4),(11-x)/(-6)),
circle(11,0,0.05),
circle(11,0,0.08),
circle(11,0,0.10)
)}}} Graph of {{{x+4y=11}}} (red) and {{{x-6y=11}}} (green)