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What does the equation $ y = 3 $ represent in $ \mathbb{R}^3 $? What does $ z = 5 $ represent? What does the pair of equations $ y = 3 $, $ z = 5 $ represent? In other words, describe the set of points $ (x, y, z) $ such that $ y = 3 $ and $ z = 5 $. Illustrate with a sketch.

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we're through representing our three. So it's gonna be a plane because artery has three coordinates and we fix one. The others who control them. Really? So it is a plane and illustrate with this catch. Good. We're going first. Draw the coordinate the O X Y Z coordinates, and why call three Crazy. Here is three zero three zero booze and the plane's gonna be like this. And here, so is parallel to O Z pain. This is why I was three. Of course you have drawn is just part of pain. And oh, what z was five pretty similar. So, uh is the axis And if you see, this is five and this playing going to be zero zero five points placing that is goingto parallel to the thanks Oh, white plane and contains this point. So of course, we're going to draw the part of the plane because it is a you cannot. You can never drawer you can average all pain. This's it was fun, okay? And asked with us the pair of equations it's white three and Z equals five represents. So if we can imagine, we have three coordinates and the two of them are fixed We can only choose one Coordinate. Really? So what does it mean, immense? We'LL have a line. Okay. When the joy it Lex was Ukraine. Okay, we have white ofthree here. We have these five here If you consider the intersects the intercepted lines just going to be well, it what? But we got to do it here because it looks kind of mass if we put them together. So we do separately. We're going to eat here. We first find we find the two points not three, not and oh, five. No. Okay, so the line that have equation wife of three. And because five, it's a line go through the point, which is lice. Vicious line on the wire was a crane. And that's a because it would be my six year. So a is with the coordinates. Zero three five. The line's gonna be through this point and parallel to the O X co ordinate So roughly is going to like this. I would do it in a in the blue one. Okay, so this is thie point a and lying. Well, go through a and parallel to the O X, the parallel to the X axis. So this is the line. In other words, describe the seven points that's wise is such that weather three end zero five. Oh, we have just explained this the seven points. Okay, that is a set. See the seven points we have this set x such that. So that's why Z now it's a set of points and the points had the standard x Y Z coordinates such that Why? Because three Z Because fi and bikes it is a room number. We do not country next. Okay, Richards, that's the banner answer the set of points.

University of Illinois at Urbana-Champaign

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