SOLUTION: How do I solve the following system of equations? {{{ x^2 + y^2 = 169}}} {{{ x^2 - 8y = 104}}} Can you provide a detailed answer, including all of the steps used to so

Algebra ->  Systems-of-equations -> SOLUTION: How do I solve the following system of equations? {{{ x^2 + y^2 = 169}}} {{{ x^2 - 8y = 104}}} Can you provide a detailed answer, including all of the steps used to so      Log On


   

Question 188802: How do I solve the following system of equations?
+x%5E2+%2B+y%5E2++=+169
+x%5E2++-+8y+=+104

Can you provide a detailed answer, including all of the steps used to solve the system of equation?

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
+x%5E2++-+8y+=+104 Start with the second equation.

+x%5E2+=+8y%2B104 Add 8y to both sides.


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+x%5E2+%2B+y%5E2++=+169 Move onto the second equation


+8y%2B104+%2B+y%5E2++=+169 Plug in +x%5E2+=+8y%2B104


+8y%2B104+%2B+y%5E2++-+169=0 Subtract 169 from both sides.


y%5E2%2B8y-65=0 Combine like terms.



Notice we have a quadratic equation in the form of ay%5E2%2Bby%2Bc where a=1, b=8, and c=-65


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-%288%29+%2B-+sqrt%28+%288%29%5E2-4%281%29%28-65%29+%29%29%2F%282%281%29%29 Plug in a=1, b=8, and c=-65


y+=+%28-8+%2B-+sqrt%28+64-4%281%29%28-65%29+%29%29%2F%282%281%29%29 Square 8 to get 64.


y+=+%28-8+%2B-+sqrt%28+64--260+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-65%29 to get -260


y+=+%28-8+%2B-+sqrt%28+64%2B260+%29%29%2F%282%281%29%29 Rewrite sqrt%2864--260%29 as sqrt%2864%2B260%29


y+=+%28-8+%2B-+sqrt%28+324+%29%29%2F%282%281%29%29 Add 64 to 260 to get 324


y+=+%28-8+%2B-+sqrt%28+324+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


y+=+%28-8+%2B-+18%29%2F%282%29 Take the square root of 324 to get 18.


y+=+%28-8+%2B+18%29%2F%282%29 or y+=+%28-8+-+18%29%2F%282%29 Break up the expression.


y+=+%2810%29%2F%282%29 or y+=++%28-26%29%2F%282%29 Combine like terms.


y+=+5 or y+=+-13 Simplify.


So the answers for "y" are y+=+5 or y+=+-13


Now simply plug these solutions into x%5E2=8y%2B104 to find the solutions for "x". I'll let you do this.


Note: you should find that there are 3 ordered pair solutions.