Algebra

Systems of Equations

System of Equations - Factorization

         

If complex numbers xx and yy satisfy the simultaneous equations x2+y2=42x23y2=67,\begin{aligned} x^2+y^2 &= 4 \\ 2x^2-3y^2 &= -67, \end{aligned} what is the value of (y+x)(yx)(y+x)(y-x)?

Let aa and bb be the values of non-zero real numbers xx and y,y, respectively, that satisfy 3x2+18y12x=9,x2+9y7x=6.3x^2+18y-12x=-9, x^2+9y-7x=-6. What is the value of a2+b2?a^2+b^2?

Let x=Ax=A and y=By=B be the solutions of the simultaneous equations 32x1+59y+1=232x159y+1=1.\begin{aligned} \frac{3}{2x-1}+\frac{59}{y+1} &= 2 \\ \frac{3}{2x-1}-\frac{59}{y+1} &= 1. \end{aligned} What is the value of BA\frac{B}{A}?

If x=a  (<0)x=a \; (< 0) and y=by=b are solutions of the simultaneous equations 2x22y2+3x56y=44,x2y2+x29y=22,2x^2-2y^2+3x-56y=44,\, x^2-y^2+x-29y=22, what is the value of bab-a?

Let x=αx=\alpha, y=βy=\beta and z=γz=\gamma be the solutions of the simultaneous equations x+y19xy=0y+z12yz=0z+x21zx=0,\begin{aligned} x+y-19xy &= 0 \\ y+z-12yz &= 0 \\ z+x-21zx &= 0, \end{aligned} where xyz0 xyz \neq 0. If the value of α+β+γ\alpha+\beta+\gamma can be expressed as ab\frac{a}{b}, where aa and bb are coprime positive integers, what is a+ba+b?

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