Lines Parallel to the Axes

Most linear equations involve both $x$ and $y$ . But two simple special cases , $x = c$ and $y = c$ , show up everywhere and confuse students unnecessarily. Each is still a linear equation in two variables, with one of $a, b$ equal to zero. Their graphs are not slanted; they are perfectly horizontal or vertical lines.

Definitions

The equation $x = c$ (for some real constant $c$ ) is shorthand for the linear equation $x - c = 0, \quad \text{i.e.} \quad 1 \cdot x + 0 \cdot y + (-c) = 0,$ in the standard form $ax + by + d = 0$ with $a = 1, b = 0, d = -c$ .

Similarly, $y = c$ is shorthand for $0 \cdot x + 1 \cdot y + (-c) = 0$ .

Concept and shapes

The graph of $x = c$ is a vertical line. Why? Every solution of $x = c$ has the form $(c, q)$ for any $q$ . All such points share the same abscissa $c$ , so they lie on a single vertical line. The line is parallel to the $y$ -axis. If $c > 0$ , it sits to the right of the $y$ -axis; if $c < 0$ , to the left; and if $c = 0$ , the line is the $y$ -axis.

The graph of $y = c$ is a horizontal line. Symmetric argument. Every solution of $y = c$ has the form $(p, c)$ for any $p$ , so all such points share the ordinate $c$ and lie on a horizontal line, parallel to the $x$ -axis. If $c > 0$ , the line sits above the $x$ -axis; if $c < 0$ , below; and if $c = 0$ , the line is the $x$ -axis.

Why no slope? Slanted lines have a slope (a rate of change of $y$ with $x$ ). Horizontal lines have slope $0$ (no change in $y$ ). Vertical lines have undefined slope (an infinite rate, sort of). We do not develop slope formally in this chapter; this hint is just orientation for next year.

Recognising these equations. The shortest description of a vertical line is just "the first coordinate is constant at $c$ ". The shortest description of a horizontal line is "the second coordinate is constant at $c$ ". On a graph, vertical lines look like the letter I; horizontal lines look like a flat dash.

Building these equations from descriptions.

"A line passes through $(3, 5)$ parallel to the $y$ -axis." Parallel to $y$ -axis means vertical, equation $x = 3$ (the abscissa of the given point).
"A line passes through $(3, 5)$ parallel to the $x$ -axis." Parallel to $x$ -axis means horizontal, equation $y = 5$ (the ordinate of the given point).

Special positions.

$x = 0$ is the $y$ -axis itself.
$y = 0$ is the $x$ -axis itself.

So the two axes are themselves graphs of linear equations.

Distance between two parallel lines.

Two vertical lines $x = c_1$ and $x = c_2$ are at distance $|c_1 - c_2|$ apart (horizontal distance).
Two horizontal lines $y = c_1$ and $y = c_2$ are at distance $|c_1 - c_2|$ apart (vertical distance).

This trivial fact is occasionally useful in geometry problems where two parallel lines bound a strip.

Crossing pattern. A vertical line $x = c$ and a horizontal line $y = d$ intersect at exactly one point: $(c, d)$ . Two distinct vertical lines do not intersect (they are parallel). Two distinct horizontal lines do not intersect.

Worked examples

Example 1. Draw the graph of $x = -2$ .

A vertical line $2$ units to the left of the $y$ -axis. Passes through $(-2, 0), (-2, 1), (-2, -1)$ , etc.

Example 2. Draw the graph of $y = 4$ .

A horizontal line $4$ units above the $x$ -axis. Passes through $(0, 4), (1, 4), (-3, 4)$ , etc.

Example 3. Find the equation of a line parallel to the $x$ -axis passing through $(-5, 7)$ .

Parallel to $x$ -axis $\Rightarrow$ horizontal $\Rightarrow$ equation $y = (\text{ordinate of the point}) = 7$ .

Example 4. Find the point where $x = 3$ and $y = -2$ intersect.

$(3, -2)$ .

Example 5. Find the distance between the lines $x = 5$ and $x = -1$ .

Both vertical. Distance $|5 - (-1)| = 6$ .

Try it yourself

Draw the graph of $x = 4$ .
Draw the graph of $y = -3$ .
Find the equation of a horizontal line through $(2, 5)$ .
Find the equation of a vertical line through $(2, 5)$ .
Where does $x = -1$ meet $y = 6$ ?
State the equation of the $y$ -axis.
State the equation of the $x$ -axis.
Distance between the lines $y = 3$ and $y = -2$ .
Is $x = -7$ parallel to the $x$ - or $y$ -axis?
The graph of $y = k$ passes through $(3, 4)$ . Find $k$ .

Pitfalls / Insight

$x = c$ is vertical, $y = c$ is horizontal. This trips up most students once. The lone variable in the equation is constant, so the other variable can vary, giving the line along the other variable's axis.
Treat axis equations as special cases. $x = 0$ and $y = 0$ are the axes themselves.
Lines parallel to the same axis are parallel to each other and never meet. No intersection point exists.

Insight. These two cases are the only "extreme" graphs in the chapter. Everything else slopes. Recognising them on sight saves time on exam questions like "where does the line $x = 2$ meet the curve $y = x + 5$ ?" , answer: at $(2, 7)$ , no computation beyond substitution.