Algebra

Systems of Equations

System of Equations - Substitution

         

If xx and yy are non-zero real numbers satisfying the system of equations

{xy=24,x22xyy=24,\begin{cases} x-y= 24, \\ x^2-2xy-y=24, \end{cases} what is the value of x+yx+y?

Let aa and bb be the values of positive integers xx and yy, respectively, that satisfy xy=5,x2+y2=73.x-y=5, x^2+y^2=73. What is the value of 5a+8b?5a+8b?

If there exists exactly one solution (x,y)(x, y) that satisfies the simultaneous equations x14y=1,x2+xy+m=0,x-14y=-1, x^2+xy+m=0, what is the value of 1m \frac{1}{m}?

For the simultaneous equations x+y=2k+16,xy+x+y=k2+5k+16x+y = 2k+16, xy+x+y = k^2+5k+16 to have real solutions, it must be the case that kabk \geq -\frac{a}{b}, where aa and bb are coprime positive integers. What is the value of a+ba+b?

For what value of kk does there exist exactly one pair (x,y)(x, y) that satisfies the simultaneous equations y14x=k,x2+y=5? y-14x=k, x^2+y=5?

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