solve the system of equations where x,y are variables and a,b are constants:
x + y = a
(x^2)*y + x*(y^2) = b
Hello, jmedsy!
Exactly where is your difficulty?
Straight substitution will solve it.
Solve the system of equations where $\displaystyle x,y$ are variables and $\displaystyle a,b$ are constants:
. . . $\displaystyle \begin{array}{cccc}x + y &= &a & [1] \\
x^2y + xy^2 &=& b & [2] \end{array}$
From [1], we have: .$\displaystyle y \:=\:a-x\;\;[3]$
Substitute into [2]: .$\displaystyle x^2(a-x) + x(a-x)^2 \:=\:b$
Expand: .$\displaystyle ax^2 - x^3 + a^2x - 2ax^2 + x^3 \:=\:b \quad\Rightarrow\quad ax^2 - a^2x + b \:=\:0$
Quadratic Formula: .$\displaystyle \boxed{x \;=\;\frac{a^2 \pm\sqrt{a^4-4ab}}{2a}}$
Substitute into [3]: .$\displaystyle y \;=\;a - \frac{a^2\pm\sqrt{a^4-4ab}}{2a} \quad\Rightarrow\quad\boxed{ y \;=\;\frac{a^2 \mp\sqrt{a^4-4ab}}{2a}} $