Question Video: Finding the Value of an Unknown in the Coordinates of a Point given That It Lies in the 𝑦𝑧-Plane Mathematics • Third Year of Secondary School

Video Transcript

Given that the point seven minus 𝑚, 12 minus 𝑚, 𝑚 lies in the 𝑦𝑧-plane, what is this point?

In this question, we’re given the coordinates of a point in three dimensions, and we can see this contains an unknown value of 𝑚. We need to determine these coordinates, which means we need to determine the value of 𝑚. And to do this, we’re told that this point lies in the 𝑦𝑧-plane. So to answer this question, we’re first going to need to recall what it means for a point to be in the 𝑦𝑧-plane. If a point lies in the 𝑦𝑧-plane, then its 𝑦- and 𝑧-coordinates can take any values. However, its 𝑥-coordinate has to be equal to zero. So, in particular, any point on the 𝑦𝑧-plane will have the form zero, 𝑏, 𝑐, where 𝑏 and 𝑐 can be any real numbers.

Therefore, because the point given to us in the question lies in the 𝑦𝑧-plane, we can conclude its 𝑥-coordinate must be equal to zero. So setting this equal to zero, we get the equation seven minus 𝑚 is equal to zero. And we can solve for the value of 𝑚. We’ll add 𝑚 to both sides of our equation, giving us that our value of 𝑚 is equal to seven. Now, all we need to do is substitute 𝑚 is equal to seven into our point. Substituting 𝑚 is equal to seven into our point, we get the point seven minus seven, 12 minus seven, seven. And if we calculate each of these expressions, we get the point zero, five, seven which is our final answer.

Therefore, given that the point seven minus 𝑚, 12 minus 𝑚, 𝑚 lies in the 𝑦𝑧-plane, we were able to determine that its 𝑥-coordinate must be equal to zero. This let us find the value of the unknown 𝑚 to be equal to seven. And we could use this to find the coordinates of our point. These coordinates were zero, five, seven.