# Question Video: Finding the Length of the Segment of the π₯-Axis Cut off by a Given Plane Mathematics

What is the length of the segment of the π₯-axis cut off by the plane 6π₯ + 3π¦ + 5π§ = 4?

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### Video Transcript

What is the length of the segment of the π₯-axis cut off by the plane six π₯ plus three π¦ plus five π§ equals four?

In this example, we have a plane, and weβre told that this plane intersects the π₯-axis of our coordinate frame. Say that intersection happens here. When our question asks what the length of the segment of the π₯-axis cut off by the plane is, itβs asking what the distances from the origin of our frame to this point of intersection. So the question is, where does this plane intersect the π₯-axis? Since weβre talking about intercepts between planes and axes, we can recall the intercept form of the equation of a plane. Written this way, the values π΄, π΅, and πΆ correspond to the π₯-, π¦-, and π§-values of the points of intersection along those respective axes.

For example, we would say that the coordinates of the point of intersection between the plane and the π₯-axis are π΄: zero, zero. If we can solve for π΄, then weβll have the answer to our question. Our task then will be to rearrange the given equation of our plane so that itβs in intercept form. Once itβs written that way, we can identify the value of π΄. Notice that the intercept form of a planeβs equation has the value of one by itself on one side. We can create a similar situation in our given plane equation by dividing both sides by four. If we do this, we get three-halves π₯ plus three-quarters π¦ plus five-fourths π§ equaling one.

And now, to force this form of our equation into intercept form, weβll write it in a bit of a strange way. We write it as π₯ divided by two-thirds plus π¦ divided by four-thirds plus π§ over four-fifths equaling one. Mathematically, these two ways of writing our planeβs equation are the same. We use this second way, though, to clarify what values are equal to π΄, π΅, and πΆ in our intercept form. πΆ is equal to four-fifths, π΅ is equal to four-thirds, and π΄, the value we want to solve for, is two-thirds. We say then the length of the segment of the π₯-axis cut off by this plane is two-thirds.