Calculus
posted by J .
Determine if the planes in each pair are parallel and distinct or coincident.
a) 5x  2y + 4z = 7
5x  2y + 4z = 3
b) 7x  3y  z = 9
21x  9y  3z = 27

The coefficients of the lefthand side determine the vector orthogonal to the plane.
Reduce each plane so that the lefthand side are identical (already are for (a)).
If the lefthand sides cannot be made identical, the planes are not parallel.
If they are parallel, look at the right hand side.
If the RHS are distinct, so are the planes. If the righthand sides are identical, the two planes are coincident. 
I don't understand it still.. So is a) parallel and b) coincidental? Please help! :(

That is correct.

Yes, you've got it right.
In (a), you see clearly that the equations are different, so they represent different planes. In addition, the coefficients of x, y and z are identical, therefore they are parallel, since the orthogonal vector of both planes are (5,2,4).
For (b), after reduction, both equations become:
7x  3y  z = 9
so that means there is only one equation that represents both planes, meaning that the two planes are coincident.
If you still have doubts, please post and explain where the doubts are, or what is not clear to you. 
Thank you! everything is clear :)

Could you please help me with this question though, I don't understand this one:
Explain why a line in threespace cannot be represented by a scalar equation.
thank you! 
Sorry, I do not understand what you mean by a scalar equation.
A line in threespace is defined by the intersection of two planes, each of which is of the form:
Ax+By+Cz+D=0
A line can also be defined in parametric form, which is basically a point to define a position in space, followed by a vector to define the direction, multiplied by a scale factor for the distance from , such as
(x,y,z)=(x0,y0,z0)+t(a,b,c)
where (a,b,c) is the direction vector, and t is a parameter. 
thank you! :)

You're welcome!