Question 246061


Start with the given system of equations:

{{{system(x+y=7,x-y=9)}}}



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-y)=(7)+(9)}}}



{{{(x+x)+(y-y)=7+9}}} Group like terms.



{{{2x+0y=16}}} Combine like terms.



{{{2x=16}}} Simplify.



{{{x=(16)/(2)}}} Divide both sides by {{{2}}} to isolate {{{x}}}.



{{{x=8}}} Reduce.



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{{{x+y=7}}} Now go back to the first equation.



{{{8+y=7}}} Plug in {{{x=8}}}.



{{{y=7-8}}} Subtract {{{8}}} from both sides.



{{{y=-1}}} Combine like terms on the right side.



So the solutions are {{{x=8}}} and {{{y=-1}}}.



Which form the ordered pair *[Tex \LARGE \left(8,-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(8,-1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-2,18,-11,9,
grid(1),
graph(500,500,-2,18,-11,9,7-x,(9-x)/(-1)),
circle(8,-1,0.05),
circle(8,-1,0.08),
circle(8,-1,0.10)
)}}} Graph of {{{x+y=7}}} (red) and {{{x-y=9}}} (green)