# Lines parallel to axes

Consider the equation \(y = 2\), or \(y - 2 = 0\). This is an equation with a single variable *y*. However, we can think of it as a two-variable linear equation in which the coefficient of *x* is 0:

\[\left( 0 \right)x + \left( 1 \right)y + \left( { - 2} \right) = 0\]

No matter what value we substitute for *x*, the value of *y* will always come out to be 0. Thus, all solutions of this two-variable linear equation are of the form \(\left( {k,2} \right)\), where *k* is some real number.

What will the graph of this linear equation look like? Well, we plot all points in the plane for which the *y*-coordinate is 2:

This is a line parallel to the *x*-axis. Thus, an equation of the form \(y = a\) represents a straight line parallel to the *x*-axis and intersecting the *y*-axis at \(\left( {0,a} \right)\).

Now, consider the equation \(x = 3\). This can also be written as a two-variable linear equation, as follows:

\[\left( 1 \right)x + \left( 0 \right)y + \left( { - 3} \right) = 0\]

Clearly, no matter what value we substitute for *y*, the value of *x* will always come out to be 3. Thus, the solutions of this equation are all of the form \(\left( {3,k} \right)\), where *k* is some real number. The graph of this equation will consist of all points whose *x*-coordinate is 3, that is, a line parallel to the *y*-axis, and passing through \(\left( {3,0} \right)\):

In general, an equation of the form \(x = a\) represents a straight line parallel to the *y*-axis and intersecting the *x*-axis at \(\left( {a,0} \right)\).

**Example 1:** What does the equation \(2x + 3 = - 1\) represent when considered as a linear equation in

I. one variable?

II. two variables?

**Solution:** I. When considered as a linear equation in one variable, this equation simply gives us one (unique) solution for *x*, which is \(x = - 2\). We can represent the solution on a (one-dimensional) number line as a single point:

II. However, when considered as a linear equation in two-variables, this represents a line parallel to the *y*-axis, as shown below:

**Example 2: **The following figure shows four lines, each of which is parallel to one of the two axes.

Determine the equation of each line.

**Solution:** \({L_1}\) is parallel to the *x*-axis and passes through \(\left( {0,2} \right)\). Thus, the equation of \({L_1}\) will be \(y = 2\).

\({L_2}\) is parallel to the *y*-axis and passes through \(\left( { - \frac{1}{2},0} \right)\). The equation of \({L_2}\) will be \(x = - \frac{1}{2}\).

Similarly, the equation of \({L_3}\) will be \(y = - \frac{3}{2}\) and that of \({L_4}\) will be \(x = \frac{5}{2}\).