Video Transcript
Write, in intercept form, the equation of the plane 16𝑥 plus two 𝑦 plus eight 𝑧 minus 16 equals zero.
In this question, we are given the equation of a plane written in general form. And we are asked to rewrite this in intercept form. We recall that the equation of the plane written in intercept form can be written 𝑥 over 𝑎 plus 𝑦 over 𝑏 plus 𝑧 over 𝑐 is equal to one, where 𝑎, 𝑏, and 𝑐 are nonzero and are the 𝑥-, 𝑦-, and 𝑧-intercepts of the plane, respectively.
One way to approach the problem is to simply rearrange our equation so it is in the correct form. Adding 16 to both sides, we have 16𝑥 plus two 𝑦 plus eight 𝑧 is equal to 16. We then divide through by 16, and this can be done term by term. The first term simplifies to 𝑥 or 𝑥 over one. The second term simplifies to 𝑦 over eight. The final term on the left-hand side is equal to 𝑧 over two. And the sum of these three terms is equal to one. The equation of the plane 16𝑥 plus two 𝑦 plus eight 𝑧 minus 16 equals zero written in intercept form is 𝑥 over one plus 𝑦 over eight plus 𝑧 over two is equal to one. As already mentioned, the values of 𝑎, 𝑏, and 𝑐, in this case one, eight, and two, are the 𝑥-, 𝑦-, and 𝑧-intercepts.
An alternative method would be to calculate these directly from the equation of the plane in general form. When the plane intersects the 𝑥-axis, we know that the 𝑦- and 𝑧-coordinates are equal to zero. Substituting these into our original equation, we have 16𝑥 minus 16 equals zero. Solving this equation for 𝑥, we have 16𝑥 equals 16 and 𝑥 equals one. Our value of 𝑎 when the equation of the plane is written in intercept form is therefore equal to one.
We can repeat this process to find the 𝑦-intercept. This time, 𝑥 is equal to zero and 𝑧 is equal to zero. We need to solve the equation two 𝑦 minus 16 equals zero. Adding 16 to both sides and then dividing through by two, we see that 𝑦 is equal to eight. This confirms the value for 𝑏 when the equation of the plane is written in intercept form. Finally, when 𝑥 is equal to zero and 𝑦 is equal to zero, 𝑧 is equal to two. This means that the 𝑦-intercept and our value of 𝑧 is equal to two. Using either one of these methods, we see that the equation of the plane written in intercept form is 𝑥 over one plus 𝑦 over eight plus 𝑧 over two equals one.
