SOLUTION: Graph both equations of eah system on the same coordinate axes. Use elimination of variables to find all pts of intersection. y = x^2 + 5x + 6 y = x + 11 Thanks

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Graph both equations of eah system on the same coordinate axes. Use elimination of variables to find all pts of intersection. y = x^2 + 5x + 6 y = x + 11 Thanks      Log On


   

Question 187799: Graph both equations of eah system on the same coordinate axes. Use elimination of variables to find all pts of intersection.

y = x^2 + 5x + 6
y = x + 11

Thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To perform elimination, simply subtract equation 2 from equation 1 to get

y-y=%28x%5E2+%2B+5x+%2B+6%29-%28x%2B11%29


0=x%5E2+%2B+5x+%2B+6-x-11


0=x%5E2%2B4x-5


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=4, and c=-5


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%284%29+%2B-+sqrt%28+%284%29%5E2-4%281%29%28-5%29+%29%29%2F%282%281%29%29 Plug in a=1, b=4, and c=-5


x+=+%28-4+%2B-+sqrt%28+16-4%281%29%28-5%29+%29%29%2F%282%281%29%29 Square 4 to get 16.


x+=+%28-4+%2B-+sqrt%28+16--20+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-5%29 to get -20


x+=+%28-4+%2B-+sqrt%28+16%2B20+%29%29%2F%282%281%29%29 Rewrite sqrt%2816--20%29 as sqrt%2816%2B20%29


x+=+%28-4+%2B-+sqrt%28+36+%29%29%2F%282%281%29%29 Add 16 to 20 to get 36


x+=+%28-4+%2B-+sqrt%28+36+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-4+%2B-+6%29%2F%282%29 Take the square root of 36 to get 6.


x+=+%28-4+%2B+6%29%2F%282%29 or x+=+%28-4+-+6%29%2F%282%29 Break up the expression.


x+=+%282%29%2F%282%29 or x+=++%28-10%29%2F%282%29 Combine like terms.


x+=+1 or x+=+-5 Simplify.


So the answers are x+=+1 or x+=+-5


Now plug these values back into either equation (I'm going to use the second)

Plug in x=1

y=x%2B11=1%2B11=12

So the first intersection is (1,12)


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Plug in x=-5

y=x%2B11=-5%2B11=6

So the second intersection is (-5,6)


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Answer:


So the solutions are

x=1 and y=12 which gives the ordered pair (1,12)

OR

x=-5 and y=6 which gives the ordered pair (-5,6)



Here's the graph to visually confirm the answer:


Graph of y+=+x%5E2+%2B+5x+%2B+6 and y=x%2B11