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Define the planes $S_1$ and $S_2$ 

$3 x-7 y-9 z\&equals;8$$$$\stackrel{\text{make plane}}{\to }$$\mathrm{<<\; Plane\; 1\; >>}$$\stackrel{\text{assign to a name}}{\to }$$S_1$

$5 x\&plus;4 y-2 z\&equals;6$$\stackrel{\text{make plane}}{\to }$$\mathrm{<<\; Plane\; 2\; >>}$$\stackrel{\text{assign to a name}}{\to }$$S_2$

Obtain the line of intersection

$S_1\&comma;S_2$ = $\mathrm{<<\; Plane\; 1\; >>}\&comma;\mathrm{<<\; Plane\; 2\; >>}$$\stackrel{\text{intersection}}{\to }$$\mathrm{<<\; Line\; 1\; >>}$$\stackrel{\text{assign to a name}}{\to }$$L$$$

Obtain the parametric equations for line $L$ 

$L$ = $\mathrm{<<\; Line\; 1\; >>}$$\stackrel{\text{representation}}{\to }$$\left[x\&equals;\frac{496}{623}\&plus;50t\&comma;y\&equals;\frac{433}{3115}-39t\&comma;z\&equals;-\frac{2279}{3115}\&plus;47t\right]$$$

$3 x-7 y-9 z\&equals;8\&comma;5 x\&plus;4 y-2 z\&equals;6$$\stackrel{\text{solve (specified)}}{\to }$$\left\{x\&equals;\frac{74}{47}\&plus;\frac{50}{47}z\&comma;y\&equals;-\frac{22}{47}-\frac{39}{47}z\right\}$

Obtain the direction vector for the line as the cross product of normals

$⟨3\&comma;-7\&comma;-9⟩\times ⟨5\&comma;4\&comma;-2⟩$ = $$

Obtain the coordinates of a point on the line of intersection

$\left[3 x-7 y-9 z\&equals;8\&comma;5 x\&plus;4 y-2 z\&equals;6\right]$

$\left[3x-7y-9z\&equals;8\&comma;5x\&plus;4y-2z\&equals;6\right]$

$\stackrel{\text{evaluate at point}}{\to }$

$\left[3x-7y\&equals;8\&comma;5x\&plus;4y\&equals;6\right]$

$\stackrel{\text{solve}}{\to }$

$\left\{x\&equals;\frac{74}{47}\&comma;y\&equals;-\frac{22}{47}\right\}$

Obtain the direction vector for $L$, which is the normal for the requisite plane

$L$ = $\mathrm{<<\; Line\; 1\; >>}$$\stackrel{\text{direction}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$N$$$

Obtain the equation of the plane

Form a sequence of the point P and the name of the normal N.

Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane

Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation

$\left[2\&comma;-3\&comma;1\right]\&comma;\mathbf{N}$$\stackrel{\text{make plane}}{\to }$$\mathrm{<<\; Plane\; 3\; >>}$$\stackrel{\text{representation}}{\to }$$50x-39y\&plus;47z\&equals;264$

Define the position vectors P and R 

$⟨2\&comma;-3\&comma;1⟩$$\stackrel{\text{assign to a name}}{\to }$$P$

$⟨x\&comma;y\&comma;z⟩$$\stackrel{\text{assign to a name}}{\to }$$R$

Implement the vector form for the equation of a plane

$\left(\mathbf{R}-\mathbf{P}\right)·\mathbf{N}\&equals;0$

$50x-264-39y\&plus;47z\&equals;0$

$L≔\mathrm{GetIntersection}\left(\mathrm{Plane}\left(3 x-7 y-9 z\&equals;8\right)\&comma;\mathrm{Plane}\left(5 x\&plus;4 y-2 z\&equals;6\right)\right)\&colon;$

$\mathrm{GetRepresentation}\left(L\&comma;\mathrm{form}\&equals;\mathrm{parametric}\right)$

$\left[x\&equals;\frac{496}{623}\&plus;50t\&comma;y\&equals;\frac{433}{3115}-39t\&comma;z\&equals;-\frac{2279}{3115}\&plus;47t\right]$

$\mathrm{GetRepresentation}\left(\mathrm{Plane}\left(\left[2\&comma;-3\&comma;1\right]\&comma;\mathrm{GetDirection}\left(L\right)\right)\right)$

$50x-39y\&plus;47z\&equals;264$