We are given two planes:

1. $x + y + z = 9$
2. $x + 5y + 5z = 9$

We are tasked with finding parametric equations for the line of intersection of these planes using the parameter $t$.

Step 1: Solve the system of equations

We will express two variables in terms of the third. Let’s subtract the first equation from the second:

$(x + 5y + 5z) - (x + y + z) = 9 - 9$

Simplifying this:

$(5y + 5z) - (y + z) = 0$

$4y + 4z = 0$

Dividing by 4:

$y + z = 0$

Thus, $z = -y$.

Step 2: Parametric equations

Now substitute $z = -y$ into the first plane equation $x + y + z = 9$:

$x + y + (-y) = 9$

Simplifying:

$x = 9$

So, the parametric equations for the line of intersection are:

$x = 9, \quad y = t, \quad z = -t$

where $t$ is the parameter.

Final Parametric Equations:

$x = 9, \quad y = t, \quad z = -t$

Would you like further details or clarifications? Here are some related questions to explore:

1. What is the geometric interpretation of the intersection of two planes?
2. How would you determine if two planes are parallel?
3. Can two planes have more than one line of intersection?
4. How do the parametric equations of a line change if we switch the roles of $y$ and $z$?
5. How would you find the angle between two intersecting planes?

Tip: When solving for the line of intersection between two planes, it's often helpful to express two variables in terms of a third, simplifying the system.