Theorem (a) (The slope- intercept equation of a plane)
Suppose that the z-intercept of a
plane is b, the slope of its vertical cross sections in the positive x -direction
is m1, and the
slope of its vertical cross sections in the positive y-direction is m2 (Figure
1). Then the plane
has the equation,
(b) (The point-slope equation of a plane ) Suppose that a
plane contains the point (x0,y0, z0),
the slope of its vertical cross sections in the positive x-direction is m1, and
the slope of its
vertical cross sections in the positive y-direction is m2 (Figure 2). Then the
plane has the
equation,
 |
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| The slope-intercept equation FIGURE 1 |
The point-slope equation FIGURE 2 |
Example 1 Give an equation of the plane with slope −6 in
the positive x-direction,
slope 7 in the positive y-direction, and z-intercept 10.
Answer: z = −6x + 7y + 10
Example 2 Give an equation of the plane through the point
(1,2,3) with slope 4 in the
positive x-direction and slope −5 in the positive y-direction.
Answer: z = 3 + 4(x − 1) − 5(y − 2)
Example 3 Find a formula for the linear function z = g(x,
y) whose values are given in
the fol lowing table .
Values of z = g(x, y)
Answer: g(x, y) = 2x − 3y + 20.
Example 4 Find a formula for the linear function z = h(x,
y) whose level curves are
given in Figure 3.
FIGURE 3
Answer: h(x, y) = 2x − 3y + 20. (Notice that h is the same
as the function g from Example 3.)
