Graphing a plane with intercepts

To graph a linear equation in three variables like $Ax + By + Cz = D$, you find its intercepts on each axis. To find the x intercept, set $y=0$ and $z=0$ and solve for $x$. Do the same for the y intercept (set $x=0, z=0$) and z intercept (set $x=0, y=0$). Plotting these three points helps visualize the plane.

For the equation $2x + 4y + z = 8$, the x intercept is $(4, 0, 0)$ because when $y=0$ and $z=0$, $2x=8$. The y intercept for the same equation is $(0, 2, 0)$, because setting $x=0$ and $z=0$ gives $4y=8$. The z intercept for $2x + 4y + z = 8$ is $(0, 0, 8)$, because if $x=0$ and $y=0$, then $z=8$.

How do you draw a giant, flat sheet (a plane) in 3D space? Just find where it pokes through each of the three axes! Find these three 'poke points' (intercepts), plot them, connect the dots to form a triangle, and you've created a neat sketch of your plane in space. It's that simple!