Question

Find the intercepts and sketch the graph of the plane. $$ 4 x+2 y+6 z=12 $$

Short answer

The intercepts of the plane are (3, 0, 0), (0, 6, 0) and (0, 0, 2). The graph is a plane that intersects these points.

Step-by-step solution

Find the x-intercept

Set y and z to zero, then solve the equation for x. This results in: \(4x = 12\), so \(x = 3\). The x-intercept is therefore the point (3, 0, 0).

Find the y-intercept

Set x and z to zero, then solve the equation for y. This leads to: \(2y = 12\), so \(y = 6\). The y-intercept is therefore the point (0, 6, 0).

Find the z-intercept

Set x and y to zero, then solve the equation for z. This gives: \(6z = 12\), so \(z = 2\). The z-intercept is therefore the point (0, 0, 2).

Sketch the graph

On a set of three-dimensional axes, plot the intercepts found in steps 1-3. Connect these points to form a triangle, then extend the edges to indicate the plane. The plane is the entire flat surface that these points define.