Problem 21 - Calculus Early Transcendentals

Question

Find the points at which the following planes intersect the coordinate axes and find equations of the lines where the planes intersect the coordinate planes. Sketch a graph of the plane. $$3 x-2 y+z=6$$

Step-by-Step Solution

Answer

The intersection points of the plane with the x, y, and z axes are (2, 0, 0), (0, -3, 0), and (0, 0, 6). The equations of the lines where the plane intersects the coordinate planes are as follows: - Intersection with the xy-plane: 3x - 2y = 6 - Intersection with the xz-plane: 3x + z = 6 - Intersection with the yz-plane: -2y + z = 6

Step 1: Intersection points with the coordinate axes

We'll find the intersection points with the x, y, and z axes by setting the other two variables to 0 and solving for the remaining variable.

Step 2: Intersection with the x-axis

Set y = 0 and z = 0 in the given equation: $$3x - 2(0) + (0) = 6 \Rightarrow x = 2$$ So the intersection point with the x-axis is (2, 0, 0).

Step 3: Intersection with the y-axis

Set x = 0 and z = 0 in the given equation: $$3(0) - 2y + (0) = 6 \Rightarrow y = -3$$ So the intersection point with the y-axis is (0, -3, 0).

Step 4: Intersection with the z-axis

Set x = 0 and y = 0 in the given equation: $$3(0) - 2(0) + z = 6 \Rightarrow z = 6$$ So the intersection point with the z-axis is (0, 0, 6).

Step 2: Equations of lines where the plane intersects the coordinate planes

To find the equations of lines where the plane intersects the coordinate planes, we'll eliminate one variable from the given equation.

Step 6: Intersection with the xy-plane

Set z = 0 in the given equation: $$3x - 2y + 0 = 6 \Rightarrow 3x - 2y = 6$$ This is the equation of the line where the plane intersects the xy-plane.

Step 7: Intersection with the xz-plane

Set y = 0 in the given equation: $$3x - 2(0) + z = 6 \Rightarrow 3x + z = 6$$ This is the equation of the line where the plane intersects the xz-plane.

Step 8: Intersection with the yz-plane

Set x = 0 in the given equation: $$3(0) - 2y + z = 6 \Rightarrow -2y + z = 6$$ This is the equation of the line where the plane intersects the yz-plane.

Step 3: Sketch a graph of the given plane

Plot the intersection points found in step 1: (2, 0, 0), (0, -3, 0), and (0, 0, 6). Connect these points to form a triangle, and extend it to create the plane. See the graph below: