Finding out which points lie on this equation?

Question

Which of the points P(3, 2, 1), Q(2, 3, 1), R(1, 4, 1) lie on the plane

$3(x − 1) + 4y − 5(z + 2) = 0?$

The equation I know is $3x + 4y + 5z.$ do I just graph this or do I plug in the points to find out the answer.

Answer 1

You can rewrite the equation $$3(x - 1) + 4y - 5(z + 2) = 0$$ to the equivalent form $$3x+4y-5z=13.$$

To find out whether some point belongs to the plane determined by this equations, you simply plug in the coordinates into the equation.

For example the point $(1,0,-2)$ belongs to plane. (If you plug into the first equation you get $3\cdot0+4\cdot0-5\cdot0=0$. If you plug into the second one, you get $3\cdot1 + 4\cdot0 - 5\cdot(-2) = 3+10=13$.

The point $(1,1,1)$ does not belong to this plane, since $3\cdot1+4\cdot1-5\cdot=3+4-5=2\ne13$.)

You can simply do the same for the given points.