Question

Find the intercepts and sketch the graph of the plane. $$ 2 x-y+z=4 $$

Short answer

The x-intercept is at (2,0,0), the y-intercept is at (0,-4,0), and the z intercept is at (0,0,4). Hence, the plane intercepts the x-axis at 2, the y-axis at -4, and the z-axis at 4.

Step-by-step solution

Find the x-intercept

The x-intercept is found by setting y and z to zero and solving the equation for x. In this case, the equation simplifies to \( 2x=4 \), so \( x=2 \). This means the x-intercept is at the point (2,0,0).

Find the y-intercept

Similarly, the y-intercept is found by setting x and z to zero and solving for y. This changes the equation to \( -y=4 \), so \( y=-4 \). This shows that the y-intercept is at the point (0,-4,0).

Find the z-intercept

To find the z-intercept, set x and y to zero, and solve for z. Here, the equation becomes \( z=4 \), so the z-intercept is at the point (0,0,4)

Sketch the plane

Now, plot the three points found (2,0,0), (0,-4,0) and (0,0,4) onto a three-dimensional coordinate system. After points are plotted, draw the plane that passes through these three points. It will result in a plane cutting the three axes at the intercepts.