Question 198021


Start with the given system of equations:

{{{system(x+y=5,x-y=-1)}}}



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



{{{(x+y)+(x-1y)=(5)+(-1)}}}



{{{(1x+1x)+(1y+-1y)=5+-1}}} Group like terms.



{{{2x+0y=4}}} Combine like terms.



{{{2x=4}}} Simplify.



{{{x=(4)/(2)}}} Divide both sides by {{{2}}} to isolate {{{x}}}.



{{{x=2}}} Reduce.



------------------------------------------------------------------



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



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



{{{2+y=5}}} Multiply.



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



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



So the solutions are {{{x=2}}} and {{{y=3}}}.



Which form the ordered pair *[Tex \LARGE \left(2,3\right)]. So the answer is the second choice.



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(2,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-8,12,-7,13,
grid(1),
graph(500,500,-8,12,-7,13,5-x,(-1-x)/(-1)),
circle(2,3,0.05),
circle(2,3,0.08),
circle(2,3,0.10)
)}}} Graph of {{{x+y=5}}} (red) and {{{x-y=-1}}} (green)