How can we find the equation of the line where two planes intersect, specifically for the planes given by q: x + y + 2z = 1 and r: x + y + z = 1?

To find the equation of the line of intersection of two planes, we start with their equations:

• Plane q: x + y + 2z = 1
• Plane r: x + y + z = 1

1. **Identifying the System of Equations**:

We have the following system:

 x + y + 2z = 1  (1)
 x + y + z = 1   (2)

2. **Subtracting the Equations**:

By subtracting equation (2) from equation (1), we eliminate x and y:

 (x + y + 2z) - (x + y + z) = 1 - 1

This simplifies to:

 z = 0

3. **Substituting Back to Find x and y**:

Substituting z = 0 back into either equation (1) or (2) gives:

 x + y + 0 = 1

Thus:

 x + y = 1

This indicates that for z = 0, x and y will satisfy the line equation of:

 y = 1 - x

4. **Parametric Equations**:

We can express the line of intersection using parameter t. Let’s set:

 x = t
 y = 1 - t
 z = 0

5. **Final Equation of the Line**:

The parametric equations for the line of intersection can be summarized as:

 x = t
 y = 1 - t
 z = 0

6. **In Vector Form**:

The line can also be expressed in vector form:

 L(t) = (t, 1 - t, 0) = (0, 1, 0) + t(1, -1, 0) 

where (0, 1, 0) is a point on the line and (1, -1, 0) is the direction vector of the line. In summary, the line of intersection of the planes q and r is:

 x = t,  
y = 1 - t,  
z = 0,  

for parameter t in the real numbers.