Chapter 5, Systems of Equations and Inequalities - Section 5.2 - Systems of Linear Equations in Several Variables - 5.2 Exercises - Page 455: 34

$x=1+7t$ $y=-1+4t$ $z=t$

Step 1. $x-2y+z=3$ $2x-5y+6z=7$ Equation 2 $+$ $(-2)$ Equation 1 $2x-3y-2z=5$ Equation 3 $+$ $(-2)$ Equation 1 Step 2. $x-2y+z=3$ Equation 1 $+$ $2$ Equation 3 $-y+4z=1$ Multiply by $-1$ $y-4z=-1$ Equation 3 $+$ Equation 2 Step 3. $x-7z=1$ $y-4z=-1$ $y-4z=-1$ Step 4. Let $z=t$. $x=1+7z=1+7t$ $y=-1+4z=-1+4t$ Step 5. Conclude the solution. The solution is $x=1+7t$, $y=-1+4t$, $z=t$.