What is the length of the segment of the 𝑦-axis cut off by the plane five 𝑥 minus four 𝑦 minus three 𝑧 plus 32 equals zero?
As we get started, let’s note that we’re told that our plane does intersect the 𝑦-axis at some point. In our sketch, let’s say that that point of intersection is here. When our question says, “What is the length of the segment of the 𝑦-axis cut off by this plane?,” it’s referring to this length here between that point of intersection and the origin. If we say that the coordinates of this point of intersection are zero, 𝐵, zero, then we can say that it’s the value of 𝐵 that corresponds to this length of interest. That’s the length of the segment of the 𝑦-axis cut off by our plane.
In our problem statement, we’re given the equation of our plane, and it’s in a form called general form. Another way to write a plane’s equation is using what’s called intercept form. In intercept form, these denominators 𝐴, 𝐵, and 𝐶 correspond to the points of intersection of our plane and the coordinate axes. Specifically, 𝐴 is the 𝑥-coordinate of the intersection point between our plane and the 𝑥-axis. 𝐶 is the 𝑧-coordinate of the intersection point between our plane and the 𝑧-axis. And 𝐵 is the 𝑦-value for that point of intersection along the 𝑦-axis. That’s what we want to solve for. And we can do this by converting the form of our plane’s equation.
Starting in the given general form, let’s write it in intercept form. We notice that in intercept form our equation will have the value of one on one side. We can start to convert our plane’s equation to that form by adding negative 32 to both sides. If we then divide both sides by negative 32, notice that on the right-hand side we get a value of one. If we distribute the negative 32 in the denominator on the left, we get five over negative 32𝑥 minus four over negative 32𝑦 minus three over negative 32𝑧 equals one. And we note that this reduces to negative five over 32 times 𝑥 plus 𝑦 over eight plus three over 32 times 𝑧.
We now have our plane’s equation written in intercept form. And we see that the value corresponding to capital 𝐵 in this general intercept form is positive eight. Therefore, the intersection point of our 𝑦-axis in plane is zero, eight, zero. And this means that the length of the segment of the 𝑦-axis this plane cuts off is eight length units.