Question 241421

Start with the given system of equations:

{{{system(x+5y=2,-6x+5y=-47)}}}



{{{-1(-6x+5y)=-1(-47)}}} Multiply the both sides of the second equation by -1.



{{{6x-5y=47}}} Distribute and multiply.



So we have the new system of equations:

{{{system(x+5y=2,6x-5y=47)}}}



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+5y)+(6x-5y)=(2)+(47)}}}



{{{(x+6x)+(5y+-5y)=2+47}}} Group like terms.



{{{7x+0y=49}}} Combine like terms.



{{{7x=49}}} Simplify.



{{{x=(49)/(7)}}} Divide both sides by {{{7}}} to isolate {{{x}}}.



{{{x=7}}} Reduce.



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{{{x+5y=2}}} Now go back to the first equation.



{{{7+5y=2}}} Plug in {{{x=7}}}.



{{{5y=2-7}}} Subtract {{{7}}} from both sides.



{{{5y=-5}}} Combine like terms on the right side.



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



{{{y=-1}}} Reduce.



So the solutions are {{{x=7}}} and {{{y=-1}}}.



Which form the ordered pair *[Tex \LARGE \left(7,-1\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(7,-1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-3,17,-11,9,
grid(1),
graph(500,500,-3,17,-11,9,(2-x)/(5),(-47+6x)/(5)),
circle(7,-1,0.05),
circle(7,-1,0.08),
circle(7,-1,0.10)
)}}} Graph of {{{x+5y=2}}} (red) and {{{-6x+5y=-47}}} (green)