To determine the equations of each plane, we analyze the position and orientation of each one with respect to the $x$, $y$, and $z$ axes. Since each plane is parallel to a coordinate plane, each will have an equation where one of the variables is held constant.

Solution:

a) First Plane:

• This plane is vertical and parallel to the $yz$-plane.
• The plane only intersects the $x$-axis, which means it has a constant $x$-value.
• Let’s assume the intersection is at $x = a$ (based on the image, it appears to intersect at $x = 3$).

Equation for (a):
$x = 3$

b) Second Plane:

• This plane is horizontal and parallel to the $xy$-plane.
• The plane intersects the $z$-axis at a specific height, so it has a constant $z$-value.
• Based on the image, it looks like $z = 2$.

Equation for (b):
$z = 2$

c) Third Plane:

• This plane is vertical and parallel to the $xz$-plane.
• The plane intersects the $y$-axis, indicating a constant $y$-value.
• Observing the image, the intersection appears to be at $y = -1$.

Equation for (c):
$y = -1$

Summary of Answers:

1. (a) $x = 3$
2. (b) $z = 2$
3. (c) $y = -1$

Would you like further explanations on these solutions or have any specific questions?

Here are five additional related questions for practice:

1. How would you find the equation of a plane if it were parallel to both the $xz$-plane and passed through $y = 5$?
2. What is the general form of a plane that is parallel to the $xy$-plane and passes through a given $z$-value?
3. If a plane intersects both the $x$-axis and the $z$-axis, but is parallel to the $yz$-plane, how would you determine its equation?
4. How can you determine if a given point lies on one of these planes?
5. What would the equation be if a plane is parallel to the $x$-axis and passes through $y = -2$ and $z = 4$?

Tip: When determining the equation of a plane that is parallel to one of the coordinate planes, identify the axis that the plane does not depend on, and set the constant value for the axis it intersects.