Equations in 3 (or more) variables
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Equations in 1 or 2 variables
From algebra, you know that a solution set of an equation in one variable is a set of points on the number line \(\mathbb{R}\). For example, the equation \(x^2 - 3x + 2 = 0\) has solution set \(\left\{1, 2\right\}\).
Similarly, a solution set of an equation in two variables is a set of points in the plane \(\mathbb{R}^2\). For example, the solutions set of the equation \(x^2 + y^2 = 1\) is the set of all points that form the circle with center \((0,0)\) and radius 1.
Sometimes the same equation can be interpreted as an equation in one variable or two variables: for example the equation \(x = 2\) as an equation in one variable has a single solution 2, while as an equation in two variables its set of solutions consists of all the points in the form \((2,y)\) for any real \(y\). These points form a vertical line.
Equations in 3 variables
Each solution of an equation in three variables is a point in the space \(\mathbb{R}^3\). For example, one of the solutions of the equation \(x + y + z = 2\) is the point \((1,1,0)\), another is \((\frac{1}{2}, \frac{1}{2}, 1)\).
As another example, consider the equation \(x^2 + y^2 + z^2 = 1\). We can rewrite the equation as \[(x - 0)^2 + (y - 0)^2 + (z - 0)^2 = 1.\]
Taking the square root of both sides will produce \[\sqrt{(x-0)^2 + (y-0)^2 + (z-0)^2} = 1.\]
Why is it OK to take the square root of both sides? How do we know that the new equation is equivalent to the old one?
The left hand side of the equation can be recognized as the distance formula \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\] with \((x_2, y_2, z_2) = (x,y,z)\) and \((x_1, y_1, z_1) = (0,0,0)\). In other words, the left hand side of the equation is the distance of the point \((x,y,z)\) from the origin, and the equation says that this distance must be equal to 1.
Therefore the solution set of the equation \(x^2 + y^2 + z^2 = 1\) is the set of all points in \(\mathbb{R}^3\) that are exactly one unit away from the origin. These points form the sphere with center in the origin and radius 1:
As another example, let’s think about the equation \(x = 2\) again, this time as an equation in three variables. Any solution of this equation must look like \((2, y, z)\), where \(y\) and \(z\) can be any real numbers (the equation does not put any requirements on them at all). The set of all such points form a plane that is parallel to the \(yz\)-plane (or, which is the same, perpendicular to the \(x\)-axis), passing through the point \((2,0,0)\):