Question

Find all values of $b$ for which $x$ and $y$ satisfying the system of equations.
$2x-y=5b,4x+y=$ $3b^2$ $-10b$. Also satisfy the inequality $x-y>3$.

Answer 1

Given equations,

$2x-y=5b\to \left(1\right)$

$4x+y=3b^2-10b\to \left(2\right)$

$\left(1\right)+\left(2\right)\Rightarrow 6x=3b^2-5b$

$x=\frac{3b^2-5b}{6}$

$\left(2\right)-2\times \left(1\right)\Rightarrow 3y=3b^2-20b$

$y=\frac{3b^2-10b}{3}$

$x-y=\frac{-3b^2+35b}{6}$

$x-y>3$

$-3b^2+35b>18$

$3b^2-35b+18<0$

$b\in (\frac{35-\sqrt{1009}}{6},\frac{35+\sqrt{1009}}{6})$