Symmetric equations for a line are derived from parametric equations for that line. Suppose, for example, we have a line in 3-space with parametric equations

x = 1 + 2t
y = 3 - 5t
z = 6 + 4t.

Solve each equation for t:

then set all the solutions equal:

.

These equations are called the symmetric equations for the line. There's potentially a problem with this sort of calculation, though - what if the normal vector has one or more zero components? We can't divide by 0, so in that case, we just use the corresponding parametric equation(s) and find symmetric equations from the remaining equations. For example, the line with parametric equations

x = x0 + at
y = y0
z = z0 + ct

will have symmetric equations

.

The line with parametric equations

x = x0
y = y0
z = z0 + ct

simply has symmetric equations x = x0, y = y0.

As with parametric equations, you can see immediately where the initial point and direction vector fit into the symmetric equations. For example, the line through the point (1, 2, 3) with direction vector [4, 5, 6] has symmetric equations

.

In 2-space, a line has just one symmetric equation, which is exactly the usual single equation for a line. For example, the symmetric equation

simplifies into 2x + 3y = -1.

Can you find a direction vector and initial point for the line with symmetric equations

?