Graph of a Linear Equation

This is the centrepiece of the chapter. An equation like $2x + 3y = 6$ describes infinitely many points, and those points fit exactly onto a single straight line in the plane. Drawing that line is the most efficient way to see a linear equation. This lesson is the technique.

Definitions

The graph of a linear equation $ax + by + c = 0$ in two variables is the set of all points $(x, y)$ in the Cartesian plane that satisfy the equation. This graph is always a straight line.

Concept and graphing technique

The headline theorem. The graph of every linear equation $ax + by + c = 0$ (where $a$ and $b$ are not both zero) is a straight line. Conversely, every straight line in the plane is the graph of some linear equation. So "linear equation" and "straight line" are the same object in two languages.

The minimal recipe. To graph a linear equation:

Find any two distinct solutions of the equation.
Plot them on the Cartesian plane.
Draw the straight line through them.

That's it. Two points determine a line, so two solutions are enough. A common practical choice: find the $x$ -intercept by setting $y = 0$ , and the $y$ -intercept by setting $x = 0$ . Those two are usually easy to compute and easy to plot.

A worked walk-through. Graph $2x + 3y = 12$ .

Set $y = 0$ : $2x = 12$ , so $x = 6$ . Point $A = (6, 0)$ .
Set $x = 0$ : $3y = 12$ , so $y = 4$ . Point $B = (0, 4)$ .

Plot $A$ and $B$ . Draw the straight line through them. That is the graph of $2x + 3y = 12$ . Every solution of the equation lies on this line; every point on this line is a solution.

Use a third point as a check. Compute one more solution and verify that it lies on the line you drew. For instance, $x = 3 \Rightarrow y = 2$ , so $(3, 2)$ should be on the line. If it is not, you have a plotting error somewhere.

When the line passes through the origin. If $c = 0$ in the standard form, the line passes through the origin. The two-intercept method then gives the same point $(0, 0)$ twice , useless. Instead, pick a non-zero value like $x = 1$ to find the second point.

Example. Graph $2x - y = 0$ . Setting $x = 0$ gives $y = 0$ , one point $(0, 0)$ . Setting $x = 1$ gives $y = 2$ , second point $(1, 2)$ . Join.

When the line is parallel to an axis. If $b = 0$ , the equation reduces to $x = c$ , a vertical line. If $a = 0$ , the equation reduces to $y = c$ , a horizontal line. These need only one number to describe and you draw them by sight; we cover these in the next lesson.

Why is the graph a straight line and not a curve? Because the equation $ax + by + c = 0$ is first degree in $x$ and $y$ . Higher-degree equations give curves: $x^2 + y^2 = r^2$ is a circle, $y = x^2$ is a parabola, $xy = 1$ is a hyperbola. The first-degree restriction is what forces straightness. A complete proof requires some work; in this chapter we take it as known.

Reading a graph backwards. Given a line on a grid, you can recover its equation by reading off two of its points and writing down the equation that passes through them. For example, a line through $(0, 3)$ and $(2, 0)$ has equation $\dfrac{x}{2} + \dfrac{y}{3} = 1$ , i.e. $3x + 2y = 6$ . Reading-and-writing graphs is a core skill.

Worked examples

Example 1. Draw the graph of $x + y = 5$ .

Intercepts: $x = 0 \Rightarrow y = 5$ , so $(0, 5)$ . $y = 0 \Rightarrow x = 5$ , so $(5, 0)$ . Plot and join with a straight line. Check: $(2, 3)$ : $2 + 3 = 5$ . Lies on the line.

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

Through origin, since $c = 0$ . Pick $x = 0 \Rightarrow y = 0$ (origin) and $x = 2 \Rightarrow y = 4$ , so $(2, 4)$ . Plot and join.

Example 3. Draw the graph of $3x - 2y = 6$ .

Intercepts: $x = 0 \Rightarrow -2y = 6 \Rightarrow y = -3$ , so $(0, -3)$ . $y = 0 \Rightarrow 3x = 6 \Rightarrow x = 2$ , so $(2, 0)$ . Plot and join. Check at $x = 4$ : $12 - 2y = 6 \Rightarrow y = 3$ , so $(4, 3)$ should lie on the line.

Example 4. Find the equation of a line that passes through $(0, 4)$ and $(2, 0)$ .

It has $y$ -intercept $4$ and $x$ -intercept $2$ . Equation: $\dfrac{x}{2} + \dfrac{y}{4} = 1$ . Multiply through by $4$ : $2x + y = 4$ .

Example 5. Graph $5x - 3y = 0$ .

Through origin: $c = 0$ . Pick $x = 3 \Rightarrow 15 - 3y = 0 \Rightarrow y = 5$ . Pick $x = 0 \Rightarrow y = 0$ . Plot $(0, 0)$ and $(3, 5)$ , join.

Try it yourself

Draw the graph of $x + y = 4$ .
Draw the graph of $2x - y = 3$ .
Draw the graph of $y = -2x + 1$ .
Find the $x$ - and $y$ -intercepts of $4x + 3y = 12$ , then draw the line.
Draw the graph of $y = 3x$ .
From the graph in question 1, read three integer solutions other than the intercepts.
Write the equation of a line passing through $(0, 0)$ and $(1, 1)$ .
Write the equation of a line passing through $(0, -2)$ and $(4, 0)$ .
Draw the graphs of $x + y = 7$ and $x - y = 1$ on the same axes. Where do they cross?
The graph of $kx - 2y = 5$ passes through $(1, -1)$ . Find $k$ .

Pitfalls / Insight

Always use at least two distinct points. A single point does not determine a line.
Plot three when you can. The third is a check; if it doesn't fall on the line, recompute.
Use a long ruler. A bumpy or curved line ruins the geometric picture.
Intercepts are the cleanest pair when both axes are crossed. If the line passes through the origin, find a different second point.

Insight. The graph of $ax + by + c = 0$ is the picture of all its solutions. Each solution is one dot; the line is the assembled dot-picture. Once you can read this picture, you can solve problems by eye , and once two lines are drawn at once, the next chapter's "pair of linear equations" becomes simply "where do these two lines cross?"