Question

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

Short answer

The intercepts are \(x = 3, y = 3, z = 3\). These intercepts define a plane in the 3D Cartesian coordinate system.

Step-by-step solution

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 \(x=3\).

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 \(y=3\).

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 \(z=3\).

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.