SOLUTION: What is the solution to the system of equations x^2 - x + 6 = y and 2x + 2y = 36? Answer 187449 by richwmiller(1774) About Me on 2010-01-09 17:12:13: You can put this solution

Algebra ->  Linear-equations -> SOLUTION: What is the solution to the system of equations x^2 - x + 6 = y and 2x + 2y = 36? Answer 187449 by richwmiller(1774) About Me on 2010-01-09 17:12:13: You can put this solution      Log On


   

Question 255158: What is the solution to the system of equations x^2 - x + 6 = y and 2x + 2y = 36?
Answer 187449 by richwmiller(1774) About Me on 2010-01-09 17:12:13:
You can put this solution on YOUR website!
x = -2 sqrt(3), y = 2 (9+sqrt(3)) x=-3.4641, y=21.4641
x = 2 sqrt(3), y = -2 (sqrt(3)-9) x=3.4641, y=14.5359
It is a line crossing a parabola.
Please explain this in detail thank you so much
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2+-+x+%2B+6+=+y+ Start with the first equation.


y+=+x%5E2+-+x+%2B+6 Rearrange the equation.


2x+%2B+2y+=+36 Move onto the second equation


2x+%2B+2%28x%5E2+-+x+%2B+6%29+=+36 Plug in y+=+x%5E2+-+x+%2B+6


2x+%2B+2x%5E2+-+2x+%2B+12+=+36 Distribute


2x%2B2x%5E2-2x%2B12-36=0 Get every term to the left side.


2x%5E2%2B0x-24=0 Combine like terms.


Notice that the quadratic 2x%5E2%2B0x-24 is in the form of Ax%5E2%2BBx%2BC where A=2, B=0, and C=-24


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%280%29+%2B-+sqrt%28+%280%29%5E2-4%282%29%28-24%29+%29%29%2F%282%282%29%29 Plug in A=2, B=0, and C=-24


x+=+%280+%2B-+sqrt%28+0-4%282%29%28-24%29+%29%29%2F%282%282%29%29 Square 0 to get 0.


x+=+%280+%2B-+sqrt%28+0--192+%29%29%2F%282%282%29%29 Multiply 4%282%29%28-24%29 to get -192


x+=+%280+%2B-+sqrt%28+0%2B192+%29%29%2F%282%282%29%29 Rewrite sqrt%280--192%29 as sqrt%280%2B192%29


x+=+%280+%2B-+sqrt%28+192+%29%29%2F%282%282%29%29 Add 0 to 192 to get 192


x+=+%280+%2B-+sqrt%28+192+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%280+%2B-+8%2Asqrt%283%29%29%2F%284%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%280%29%2F%284%29+%2B-+%288%2Asqrt%283%29%29%2F%284%29 Break up the fraction.


x+=+%22%22+%2B-+2%2Asqrt%283%29 Reduce.


x+=+2%2Asqrt%283%29 or x+=+-2%2Asqrt%283%29 Break up the expression.


So the solutions in terms of 'x' are x+=+2%2Asqrt%283%29 or x+=+-2%2Asqrt%283%29


which approximate to x=3.464 or x=-3.464


Now let's find the corresponding solution for y when x=2%2Asqrt%283%29


y+=+x%5E2+-+x+%2B+6 Start with the given equation.


y+=+%282%2Asqrt%283%29%29%5E2+-+2%2Asqrt%283%29+%2B+6 Plug in x=2%2Asqrt%283%29


y+=+12+-+2%2Asqrt%283%29+%2B+6 Square 2%2Asqrt%283%29 to get %282%2Asqrt%283%29%29%5E2=2%5E2%2A%28sqrt%283%29%29%5E2=4%2A3=12


y+=+18+-+2%2Asqrt%283%29 Combine like terms.


So when x=2%2Asqrt%283%29, y+=+18+-+2%2Asqrt%283%29 which approximates to y=14.536



Now let's find the corresponding solution for y when x=-2%2Asqrt%283%29


y+=+x%5E2+-+x+%2B+6 Start with the given equation.


y+=+%28-2%2Asqrt%283%29%29%5E2+%2B+2%2Asqrt%283%29+%2B+6 Plug in x=-2%2Asqrt%283%29


y+=+12+%2B+2%2Asqrt%283%29+%2B+6 Square -2%2Asqrt%283%29 to get %28-2%2Asqrt%283%29%29%5E2=%28-2%29%5E2%2A%28sqrt%283%29%29%5E2=4%2A3=12


y+=+18+%2B+2%2Asqrt%283%29 Combine like terms.


So when x=-2%2Asqrt%283%29, y+=+18+-+2%2Asqrt%283%29 which approximates to y=21.464