SOLUTION: Hi! Use the substitution method to find all solutions of the system of equations. (Round each answer to two decimal places.) {2x^2+4y=11 x^2-y^2=1/2 thanks for the homew

Algebra ->  Systems-of-equations -> SOLUTION: Hi! Use the substitution method to find all solutions of the system of equations. (Round each answer to two decimal places.) {2x^2+4y=11 x^2-y^2=1/2 thanks for the homew      Log On


   

Question 200162: Hi!
Use the substitution method to find all solutions of the system of equations. (Round each answer to two decimal places.)
{2x^2+4y=11
x^2-y^2=1/2
thanks for the homework help!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2-y%5E2=1%2F2 Start with the second equation.


x%5E2=1%2F2%2By%5E2 Add y%5E2 to both sides.


2x%5E2%2B4y=11 Move back to the first equation.


2%281%2F2%2By%5E2%29%2B4y=11 Plug in x%5E2=1%2F2%2By%5E2


2%2B2y%5E2%2B4y=11 Distribute


2%2B2y%5E2%2B4y-11=0 Subtract 11 from both sides.


2y%5E2%2B4y-9=0 Combine like terms.


Notice that the quadratic 2y%5E2%2B4y-9 is in the form of Ay%5E2%2BBy%2BC where A=2, B=4, and C=-9


Let's use the quadratic formula to solve for "y":


y+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


y+=+%28-%284%29+%2B-+sqrt%28+%284%29%5E2-4%282%29%28-9%29+%29%29%2F%282%282%29%29 Plug in A=2, B=4, and C=-9


y+=+%28-4+%2B-+sqrt%28+16-4%282%29%28-9%29+%29%29%2F%282%282%29%29 Square 4 to get 16.


y+=+%28-4+%2B-+sqrt%28+16--72+%29%29%2F%282%282%29%29 Multiply 4%282%29%28-9%29 to get -72


y+=+%28-4+%2B-+sqrt%28+16%2B72+%29%29%2F%282%282%29%29 Rewrite sqrt%2816--72%29 as sqrt%2816%2B72%29


y+=+%28-4+%2B-+sqrt%28+88+%29%29%2F%282%282%29%29 Add 16 to 72 to get 88


y+=+%28-4+%2B-+sqrt%28+88+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


y+=+%28-4+%2B-+2%2Asqrt%2822%29%29%2F%284%29 Simplify the square root


y+=+%28-4%2B2%2Asqrt%2822%29%29%2F%284%29 or y+=+%28-4-2%2Asqrt%2822%29%29%2F%284%29 Break up the expression.


y+=+%28-2%2Bsqrt%2822%29%29%2F%282%29 or y+=+%28-2-sqrt%2822%29%29%2F%282%29 Reduce


So the solutions (for "y") are y+=+%28-2%2Bsqrt%2822%29%29%2F%282%29 or y+=+%28-2-sqrt%2822%29%29%2F%282%29


which approximate to y=1.35 or y=-3.35


Now plug these values into x%5E2=1%2F2%2By%5E2 to find the corresponding "x" values.

Note: You should get four ordered pair solutions (there are two x values for each y value you just found).