Problem 4
Question
Find the intercepts and sketch the graph of the plane. $$ x+y+z=3 $$
Step-by-Step Solution
Verified Answer
The intercepts are \(x = 3, y = 3, z = 3\). These intercepts define a plane in the 3D Cartesian coordinate system.
1Step 1: Find the x-intercept
To find the x-intercept, set y and z as zero in the equation \(x+y+z=3\). This simplifies to \(x+0+0=3\), or \(x=3\). So the x-intercept is \(x=3\).
2Step 2: Find the y-intercept
To find the y-intercept, set x and z as zero in equation \(x+y+z=3\). This simplifies to \(0+y+0=3\), or \(y=3\). So the y-intercept is \(y=3\).
3Step 3: Find z-intercept
To find the z-intercept, set x and y as zero in equation \(x+y+z=3\). This simplifies to \(0+0+z=3\), or \(z=3\). So the z-intercept is \(z=3\).
4Step 4: Plot the intercepts
Now, plot these three intercepts on the Cartesian coordinate system. The intercepts are points (3,0,0), (0,3,0), and (0,0,3). Connect these points and fill in the plane bounded by these points to sketch the graph of the plane.
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