 In two dimensions, an equation in x and y defines a one-dimensional curve. For example, 3x plus 5y equals 15, well, that's a straight line. x squared plus y squared equals 16, well, that's a circle. x cubed plus y cubed equals x squared plus y squared to the fourth times. xy is a curve of some sort. Now, if we go up to three dimensions, an equation in x, y, and z is going to define a two-dimensional surface. And we've seen a couple of these. z equals x squared plus y squared is a paraboloid. z squared equals x squared plus y squared is a cone. z cubed plus x squared y squared plus 8xy equals 100 is a surface of some sort. And so, given an equation, we should be able to use level curves to sketch the surface. And we should also be able to go backwards. Given a surface, we should be able to write an equation. So, for example, let's say we want to find the equation of the sphere of radius 4 centered at the origin. So, here it's helpful to remember that a sphere consists of all points a specific distance from its center. Now, the sphere is centered at the origin, so we want the distance from any point x, y, z on the sphere to the origin to be equal to 4. Now, the Pythagorean theorem holds in three dimensions, so we could use the distance formula. The distance between the two points is the square root of the sum of the squares of the differences. And we want that equal to 4. And we can simplify this. And we have a square root, and we can square both sides to eliminate the square root to get a nicer looking equation. Well, what if we try something like a plane? Well, we've already found some equations for planes. z equals 0 is the equation for the x, y plane. y equals 0 is the equation for the x, z plane. And x equals 0 is the equation for the y, z plane. And we've also graphed some of these. z equals 5 is a plane. 3x plus 2y equals 12 is a plane. 2x plus 3y plus 5z equals 30. Well, when we graphed it, we saw that it looks something like this, which also suggests that it's a plane. And this suggests the following. The graph of ax plus by plus cz equals d is a plane. So let's find some equations. Well, you know from geometry that we need three points to specify a plane. So let's find the equation of the plane through these three points. So the equation of a plane is ax plus by plus cz equals d. Now, the important thing to remember is we don't know a, b, c or d. So these are our variables. And we can solve for them by setting up a system of equations. So first, remember, if a point is on a surface, it makes the equation for the surface true. And since 2, 1, 5 is on the plane, then if I let x equals 2, y equals 1, z equals 5, I should get a true statement. So substituting in those values for x, y, and z. And so we know that 2a plus b plus 5c must equal d. Well, that worked well enough. Let's take a look at our second point. We know that 1, 4, negative 3 is on the plane. And so that gives us the equation a plus 4b minus 3c equal to d. And again, 1, 2, negative 4 is on the plane. And this will give us an equation. And now we have a system of equations. And we can solve this system of equations. First, we'll reorder the equations. We don't really need to. It's just convenient if we do so. And as we'll see in a moment, it turns out to be very convenient if our leading variable has coefficient 1. So we'll make one of these two equations our first equation. And then the others can be in any order that we want. So let's try to solve the system of equations. And so notice that the first two equations have the same coefficient on a. So if we subtract the first from the second, we'll eliminate that a. And we don't need that second equation anymore. The third equation has coefficient of a twice the first. So if we subtract twice the first equation from the third, we can eliminate the variable a. And that gives us a new third equation. And again, we no longer really need that original third equation. Now, in our second and third equations, the coefficients of b are different. But if we multiply the second equation by 3 and the third equation by 2 and add, we eliminate b and we get a new third equation. Now, since we have three equations, but four unknowns, we can try to pick a value for the fourth unknown and compute the others. So notice that if we choose a value for d, then finding c will require dividing by 29 and finding b will require dividing by 2. So for convenience, we might choose d to equal 29 times 2. That way, when we have to divide by 29 or by 2, at least we're starting with something that can be divided by both and hopefully will avoid fractions. So if d is 29 times 2, then our last equation becomes and we solve. And so that gives us d equals 29 times 2, 58, and c equals negative 4. So now that I know c and d, we can substitute into our third equation and get and solve. And now that we know b, c, and d, we can substitute into our first equation and get our value for a, b, c, and d, and so our equation will be. And you might notice that each coefficient has a factor of 2, so we could reduce this equation, but we won't bother. What if we wanted to write the equation of a line? Well, let's make a few observations. Notice that in two dimensions, a one-dimensional straight line has an equation with two variables, the line 3x plus 5y equals 60. But a zero-dimensional point requires two equations, the point where x is equal to 5, y is equal to 8. And that's because the point is at the intersection of two lines. And in fact, if you think about it, x equals 5, y equals 8 are the equations for two lines. Now, when we go up to three dimensions, in three dimensions, a two-dimensional plane has an equation with three variables. But if we drop down and try to consider a one-dimensional line, this is going to require two equations. And if you go back to your geometry, if you intersect two planes, you'll get a straight line. And so if we want to talk about lines, we have to say the line at the intersection of two planes where we give the equations for the two planes. And what this means is if we wanted to find the equation for a line through two points, we'd need to find the equations for two planes. Now, we could do this, but maybe there's an easier way. And in order to get there, we'll need to introduce some more ideas first. So we'll take a look at those next.