Question

In Exercises \(63-70,\) label any intercepts and sketch a graph of the plane. $$ 4 x+2 y+6 z=12 $$

Short answer

The x-intercept of the plane is (3,0,0), the y-intercept is (0,6,0) and the z-intercept is (0,0,2). The graph of the plane is drawn by plotting the intercepts and forming a triangle from the origin to these three points, where the plane contains this triangle.

Step-by-step solution

Finding the x-intercept

Set y=z=0 in the given equation, that is \(4x + 2(0) + 6(0) = 12\), which simplifies to \(4x = 12\). Solve this equation to determine the x-intercept. The solution is \(x = 12/4 = 3\). Thus the x-intercept is (3,0,0).

Finding the y-intercept

Similarly, set x=z=0 in the given equation, that is \(4(0) + 2y + 6(0) = 12\), which simplifies to \(2y = 12\). Solve this equation to determine the y-intercept. The solution is \(y = 12/2 = 6\). Thus the y-intercept is (0,6,0).

Finding the z-intercept

Finally, set x=y=0 in the given equation to get to \(4(0) + 2(0) + 6z = 12\), which simplifies to \(6z = 12\). Solve this equation to determine the z-intercept. The solution is \(z = 12/6 = 2\). Thus the z-intercept is (0,0,2).

Graphing the plane

Having the intercepts, one can proceed to draw a 3D axes system. The points (3,0,0), (0,6,0), and (0,0,2) are plotted and a triangle is formed from the origin to these three points. The plane will be the one that contains this triangle.