Problem 2
Question
Find the intercepts and sketch the graph of the plane. $$ 3 x+6 y+2 z=6 $$
Step-by-Step Solution
Verified 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).
1Step 1: 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.
2Step 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.
3Step 3: 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.
4Step 4: 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.
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