How would you find the Cartesian Equation that fits the following requirements?

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The Cartesian equation must pass through the point $(3,-1,2)$ and must be perpendicular to the LOI of the planes $3x-2y+1=0$ and $3x+4z-5=0$ Vectors was never really my strong suit, if anyone could help me that would be great.

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The vectors normal to your planes are $(3,-2,0)$ and $(3,0,4)$ Find their cross product $$ (3,-2,0) \times (3,0,4) = (8,-12,6)$$ which is a vector parallel to the LOI . Using the given point the LOI is $$(x,y,z)= (3,-1,2)+t(8,-12,6)$$

To convert to a Cartesian equation eliminate $t$ in the system of equations $$\begin{cases} x=3+8t & \\ y=-1-12t& \\ z=2+6t & \end{cases}$$