SOLUTION: Help! Im stuck on this equation: {{{ x^2+y^2+x-y-6=0 }}} {{{ 2x^2+2y^2+3x+y-12=0 }}} (note that these two equations are in the same question) Please help!

(1)  First step is to multiply first equation by 2.



(2)  Second step is to subtract the second equation from the modified first equation.

     By doing so, you will cancel quadratic terms and will obtain a linear equation.



(3)  Thus after first two steps, you will reduce the original system of two equations each of the degree 2,

     to a SIMPLER EQUIVALENT system of two equations, one of which is of the degree 2 and the other is linear.



(4)  From the linear equation, express one unknown via another and substitute it into the other equation of the degree 2.

     By making this substitution, you will reduce the problem to the solution of a quadratic equation for only one single unknown.


     Solve this equation and find the values of this unknown.



(5)  Then substitute these values into the linear equation and find two values of the other unknown.



(6)  The given system represents two circles in coordinate plane, that presumably intersect in two points.

     The modified system represents a circle and a straight line, which presumably intersect at the same two pints.

     The solution which you will find algebraically,  will represent these two intersection points.



(7)  Still there is an opportunity that the quadratic equation has NO real solutions, or has ONLY ONE real solution,

     instead of two.

     It will mean that the two original circles do not intersect; or do touch in one single point only.

     I don't know in advance, what will happen --- YOU should find out, what will happen, actually.