Question 365716


Start with the given system of equations:

{{{system(x+y=8,-x+2y=7)}}}



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)+(-1x+2y)=(8)+(7)}}}



{{{(1x+-1x)+(1y+2y)=8+7}}} Group like terms.



{{{0x+3y=15}}} Combine like terms.



{{{3y=15}}} Simplify.



{{{y=(15)/(3)}}} Divide both sides by {{{3}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



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{{{x+y=8}}} Now go back to the first equation.



{{{x+5=8}}} Plug in {{{y=5}}}.



{{{x=8-5}}} Subtract {{{5}}} from both sides.



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



So the solutions are {{{x=3}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(3,5\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(3,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-7,13,-5,15,
grid(1),
graph(500,500,-7,13,-5,15,8-x,(7+x)/(2)),
circle(3,5,0.05),
circle(3,5,0.08),
circle(3,5,0.10)
)}}} Graph of {{{x+y=8}}} (red) and {{{-x+2y=7}}} (green) 



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Jim