Question

Find the intercepts and sketch the graph of the plane. $$ 3 x+3 y+5 z=15 $$

Short answer

The x-intercept is 5, the y-intercept is 5, and the z-intercept is 3. Plot these points on a 3D graph and connect them to sketch the plane.

Step-by-step solution

Find the x-intercept

To find the x-intercept, set y and z to zero and solve for x in the given equation: \( 0x + 3(0) + 5(0) = 15 \). Solving this, we find that x = 5.

Find the y-intercept

To find the y-intercept, we will do the same as in step 1, but this time set x and z to zero and solve for y in the equation: \( 3(0) + 3y + 5(0) = 15 \). Solving for y, we find that y = 5 as well.

Find the z-intercept

To find the z-intercept, set x and y to zero and solve for z using the equation \( 3(0) + 3(0) + 5z = 15 \). Solving this, we find that z = 3.

Plot the intercepts and sketch the plane

Now that we have the x, y, and z intercepts, we can plot these points on a 3D graph and connect them to form the plane. The x, y and z intercepts correspond to the points (5, 0, 0), (0, 5, 0) and (0, 0, 3). Sketch the plane going through these three points.