SOLUTION: Hi! Use the elimination method to find all solutions of the system of equations. (Round each answer to two decimal places. Enter NONE in any unused answer blanks.) {x^2-y^2

Algebra ->  Systems-of-equations -> SOLUTION: Hi! Use the elimination method to find all solutions of the system of equations. (Round each answer to two decimal places. Enter NONE in any unused answer blanks.) {x^2-y^2      Log On


   

Question 200163: Hi!

Use the elimination method to find all solutions of the system of equations. (Round each answer to two decimal places. Enter NONE in any unused answer blanks.)
{x^2-y^2=25
2x^2-y^2=x+55
thanks for the help!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the given system
system%28x%5E2-y%5E2=25%2C2x%5E2-y%5E2=x%2B55%29


Multiply both sides of the first equation by -1

system%28-1%28x%5E2-y%5E2%29=-1%2825%29%2C2x%5E2-y%5E2=x%2B55%29


Distribute and multiply

system%28-x%5E2%2By%5E2=-25%2C2x%5E2-y%5E2=x%2B55%29



Now add the two equations


%28-x%5E2%2By%5E2%29%2B%282x%5E2-y%5E2%29=-25%2Bx%2B55


%28-x%5E2%2B2x%5E2%29%2B%28y%5E2-y%5E2%29=x%2B%28-25%2B55%29 Group like terms.


x%5E2=x%2B30 Combine like terms.


x%5E2-x-30=0 Get all terms to the left side.


Notice that the quadratic x%5E2-x-30 is in the form of Ax%5E2%2BBx%2BC where A=1, B=-1, and C=-30


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-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-30%29+%29%29%2F%282%281%29%29 Plug in A=1, B=-1, and C=-30


x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-30%29+%29%29%2F%282%281%29%29 Negate -1 to get 1.


x+=+%281+%2B-+sqrt%28+1-4%281%29%28-30%29+%29%29%2F%282%281%29%29 Square -1 to get 1.


x+=+%281+%2B-+sqrt%28+1--120+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-30%29 to get -120


x+=+%281+%2B-+sqrt%28+1%2B120+%29%29%2F%282%281%29%29 Rewrite sqrt%281--120%29 as sqrt%281%2B120%29


x+=+%281+%2B-+sqrt%28+121+%29%29%2F%282%281%29%29 Add 1 to 120 to get 121


x+=+%281+%2B-+sqrt%28+121+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%281+%2B-+11%29%2F%282%29 Take the square root of 121 to get 11.


x+=+%281+%2B+11%29%2F%282%29 or x+=+%281+-+11%29%2F%282%29 Break up the expression.


x+=+%2812%29%2F%282%29 or x+=++%28-10%29%2F%282%29 Combine like terms.


x+=+6 or x+=+-5 Simplify.


So the solutions (for the x values) are x+=+6 or x+=+-5


Now take each x value and plug them into either equation to solve for "y". Let me know if you need more help from here.


Note: In this case, you should get 4 ordered. However, one solution is repeated twice. So there are really only 3 ordered pair solutions.