Question

Find the intercepts and sketch the graph of the plane. $$ 3 x+6 y+2 z=6 $$

Short answer

The X-intercept is 2, the Y-intercept is 1 and the Z-intercept is 3. The graph of the plane is a triangular plane passing through the points (2, 0, 0), (0, 1, 0), and (0, 0, 3).

Step-by-step solution

Find the X-intercept

To find the X-intercept, set y = 0 and z = 0 in the given equation. This will give: 3x + 6(0) + 2(0) = 6 => 3x = 6 => x = 2. So, the X-intercept is 2.

Find the Y-intercept

To find the Y-intercept, set x = 0 and z = 0 in the given equation. This will give: 3(0) + 6y + 2(0) = 6 => 6y = 6 => y = 1. So, the Y-intercept is 1.

Find the Z-intercept

To find the Z-intercept, set x = 0 and y = 0 in the given equation. This will give: 3(0) + 6(0) + 2z = 6 => 2z = 6 => z = 3. So, the Z-intercept is 3.

Sketch the Graph of the Plane

Use the intercepts to sketch the graph of the plane. The intercepts represent the points where the plane crosses the X, Y, and Z axes. Therefore, draw a set of 3D axes, and mark the points (2,0,0), (0,1,0), and (0,0,3) on the X, Y, and Z axes respectively. Then, connect these points with straight lines to form a triangular plane. This represents the graph of the given plane equation.