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Understanding linear equations in three variables is crucial when trying to sketch planes in 3D space. A linear equation in the form of ax + by + cz = d, where a, b, and c are coefficients and d is a constant, defines a flat surface or a plane in three-dimensional space. It's a relationship that shows how x, y, and z variables affect each other linearly. This means that if you were to graph the relationship, it would result in a straight plane with no curves. The plane can be visualized by plotting points that satisfy the equation and then connecting these points.

For instance, our exercise involves the equation 2x - z = 2, which can be considered a linear equation with three variables where the coefficient of y is zero. This implies that the plane is parallel to the y-axis. To sketch it, we only need to consider how x and z interact, since any value for y is accepted by the equation. This plane cuts the x-axis and the z-axis at certain points which we refer to as intercepts.

The x-intercept of a plane in 3D space is the point where the plane crosses the x-axis. To find this point, the other variables—in our case, y and z, must be set to zero, and we solve the equation for x. This intercept offers insight into the plane's position relative to the x-axis.

In our example, with the equation 2x - z = 2, setting z = 0 gives us the solution x = 1. This tells us that the x-intercept of the plane is the point (1,0,0), meaning the plane intersects the x-axis one unit away from the origin. This point will be one of the guides to sketch the plane, offering a clear view of how the plane stretches in 3D space.

Similarly, the z-intercept can be found by setting the other two variables to zero. The z-intercept is where the plane crosses the z-axis. For our specified equation, 2x - z = 2, setting x = 0 and solving for z gives us z = -2. The z-intercept is the point (0,0,-2) and tells us the point of intersection between the plane and the z-axis below the origin.

Identifying the z-intercept is vital for sketching the plane since it completes the set of intercepts needed to outline the plane. It provides a clear direction of the plane's tilt or angle in relation to the z-axis and helps in constructing the plane's orientation in three-dimensional space alongside the x-intercept.