### Video Transcript

Which of the following points lies in the plane three times quantity 𝑥 plus four minus two times quantity 𝑦 plus one minus seven times quantity 𝑧 minus six equals zero? (A) Three, negative two, negative seven. (B) Seven, negative one, negative 13. (C) Four, one, negative six. (D) Negative four, negative one, six.

Okay, the way to figure out which of these points lies in the given plane is to substitute in the 𝑥-, 𝑦-, and 𝑧-values given in these points into the plane equation. When we do that and calculate the left-hand side of this equation, if it equals zero, then that point does lie in the plane. What we’ll do then is substitute in these points into our given plane equation one by one starting with option (A).

Doing this, we have three times the quantity three plus four minus two times the quantity negative two plus one minus seven times the quantity negative seven minus six. This equals three times seven minus two times negative one minus seven times negative 13. That’s 21 plus two plus 91, or 114. Since this result is not equal to zero as the right-hand side of our plane equation has, then we can say that a point with the coordinates three, negative two, negative seven does not lie in this plane. Option (A) then is eliminated from consideration.

Now let’s look at the point in option (B). We have three times the quantity seven plus four minus two times the quantity negative one plus one minus seven times the quantity negative 13 minus six. That’s three times 11 minus two times zero minus seven times negative 19. That’s 33 plus 133, or 166, also not zero. We can see then that this point seven, negative one, negative 13 is also not in the plane.

Moving on to option (C), three times the quantity four plus four minus two times the quantity one plus one minus seven times the quantity negative six minus six equals three times eight minus two times two minus seven times negative 12. This is 24 minus four plus 84, or 104. For this point too then, when we plug it into our planes equation, we don’t get zero. The point doesn’t lie in the plane.

Let’s hope we find a different result for option (D). Three times the quantity negative four plus four minus two times the quantity negative one plus one minus seven times the quantity six minus six equals three times zero minus two times zero minus seven times zero. This all adds up to zero. And so we found evidence that this point negative four, negative one, six does lie in the given plane. And so that’s our answer. The point negative four, negative one, six lies in the plane three times the quantity 𝑥 plus four minus two times the quantity 𝑦 plus one minus seven times the quantity 𝑧 minus six equals zero.