The distance between the two parallel planes is
units.
The given parallel planes are represented by the equations:
1. 2z = 6y - 4x
2. 3z = 2 - 6x + 9y
To find the distance between these parallel planes, we can use the formula:
![\[ \text{Distance} = (\left|D_2 - D_1\right|)/(√(A^2 + B^2 + C^2)) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/h3tgewro0ggdntktatg9mkz3z00gga0z82.png)
where the equations of the planes are in the form Ax + By + Cz = D.
Let's rewrite the given equations in the form Ax + By + Cz = D:
1.

2.

Now, we can use the coefficients \(A\), \(B\), \(C\), and \(D\) in the formula.
For Plane 1:
![\[ A_1 = 4, \quad B_1 = -6, \quad C_1 = 2, \quad D_1 = 0 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/iz6k3h4rrsmx0qs2wpn8rnfbzysx4e6j4h.png)
For Plane 2:
![\[ A_2 = 6, \quad B_2 = -9, \quad C_2 = 3, \quad D_2 = 2 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/o7zr5nm95t7ut5xq5tkhduqv288l1d20w1.png)
Now, plug these values into the formula:
![\text{Distance} = (\left|2 - 0\right|)/(√(4^2 + (-6)^2 + 2^2)) \]\\\text{Distance} = (2)/(√(16 + 36 + 4)) \]\\\text{Distance} = (2)/(√(56)) \]\\ \text{Distance} = (2)/(2√(14)) \]\\ \text{Distance} = (1)/(√(14)) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/n0k7ikjnzm3rii2q0m0jot52rifc3rjvk9.png)
So, the distance between the given parallel planes is
.