UTM Ideals Varieties and Algorithm - Chapter 1 Section 3 Exercise 1

Problem:

Solution:

Take the sum of the the equations, we have:

$\begin{eqnarray*} 2x + 3y - z &=& 1 \\ z &=& 2x + 3y - 1 \\ w &=& x + y + z - 2 \\ &=& x + y + (2x + 3y - 1) + 2 \\ &=& 3x + 4y + 1 \end{eqnarray*}$

Therefore the solutions are, in parameters $s$ and $t$, are

$\left(\begin{array}{c}x \\ y \\ z \\ w \end{array}\right) = \left(\begin{array}{c}s \\ t \\ 2s + 3t - 1 \\ 3s + 4t + 1 \end{array}\right)$