Graph Linear Equations in Two Variables

Recognize the relationship between the solutions of an equation and its graph

In the previous section, we found several solutions to the equation $3x + 2y = 6$. They are listed in the table below. So the ordered pairs $(0, 3)$, $(2, 0)$, and $\left(1, \tfrac{3}{2}\right)$ are some solutions to the equation $3x + 2y = 6$. We can plot these solutions in the rectangular coordinate system, as shown below.

Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation $3x + 2y = 6$, shown below. Notice the arrows on the ends of the line — they indicate the line continues in both directions.

Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions.

Notice that the point $(-2, 6)$ is on the line shown below. If you substitute $x = -2$ and $y = 6$ into the equation, you find that it is a solution to the equation:

So the point $(-2, 6)$ is a solution to the equation $3x + 2y = 6$. What about $(4, 1)$?

So $(4, 1)$ is not a solution to the equation $3x + 2y = 6$. Therefore, the point $(4, 1)$ is not on the line. This is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation, and every solution of this equation is a point on this line. This line is called the graph of the equation $3x + 2y = 6$.

Example. The graph of $y = 2x - 3$ is shown below.

For each ordered pair, decide: (a) is the ordered pair a solution to the equation? (b) is the point on the line? We’ll check $A(0, -3)$, $B(3, 3)$, $C(2, -3)$, and $D(-1, -5)$.

(a) Substitute the $x$- and $y$-values into the equation $y = 2x - 3$ to check whether each ordered pair is a solution.

So $(0, -3)$, $(3, 3)$, and $(-1, -5)$ are solutions, while $(2, -3)$ is not a solution.

(b) Plotting the four points confirms it: $(0, -3)$, $(3, 3)$, and $(-1, -5)$ fall on the line $y = 2x - 3$, but $(2, -3)$ does not. The points that are solutions to $y = 2x - 3$ are on the line, and the point that is not a solution is not on the line.

Graph a linear equation by plotting points

There are several methods that can be used to graph a linear equation. The method we used to graph $3x + 2y = 6$ is called plotting points, or the Point-Plotting Method.

It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line, but it will be the wrong line. If you use three points and one is incorrect, the points will not line up — this tells you something is wrong and you need to check your work.

When an equation includes a fraction as the coefficient of $x$, we can still substitute any numbers for $x$. But the math is easier if we make “good” choices for the values of $x$ — choosing multiples of the denominator avoids fraction answers, which are hard to graph precisely.

Example. Graph the equation $y = \tfrac{1}{2}x + 3$.

Since this equation has the fraction $\tfrac{1}{2}$ as a coefficient of $x$, we choose values of $x$ carefully. We use zero as one choice and multiples of $2$ for the other choices, so that $\tfrac{1}{2}x$ comes out even each time.

Plotting the points, checking that they line up, and drawing the line gives the graph of $y = \tfrac{1}{2}x + 3$.

So far, all the equations we graphed had $y$ given in terms of $x$. Now we’ll graph an equation with $x$ and $y$ on the same side, such as $2x + y = 3$. When an equation is in this form, it is often easier to find the intercepts than to pick three arbitrary values — first solve the equation for $y$ so it is easier to find points, or find one point by letting $x = 0$ and another by letting $y = 0$.

If we solve $2x + y = 3$ for $y$, we get $y = -2x + 3$. The solutions for $x = 0$, $x = 1$, and $x = -1$ are shown below.

Example. Graph the equation $2x - 3y = 6$.

Since both $x$ and $y$ are on the same side, it is not that easy to solve for $y$ in one step, so we leave the equation in standard form and find a first point by letting $x = 0$, a second point by letting $y = 0$, and a third point by choosing some other value.

We need a third point. We’ll let $x = 6$:

Plot the points, check that they line up, and draw the line. If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answer key? If the points where the graphs cross the $x$- and $y$-axes are the same, the graphs match!

Graph vertical and horizontal lines

Can we graph an equation with only one variable — just $x$ and no $y$, or just $y$ without an $x$? How will we make a table of values to get the points to plot?

Let’s consider the equation $x = -3$. This equation has only one variable, $x$. The equation says that $x$ is always equal to $-3$, so its value does not depend on $y$. To make a table of values, we write $-3$ in for all the $x$-values, then choose any values for $y$ — we’ll use $1$, $2$, and $3$.

Plotting these points and connecting them with a straight line gives a vertical line.

Example. Graph the equation $x = 2$.

The equation has only one variable, $x$, and $x$ is always equal to $2$. We create a table where $x$ is always $2$, then put in any values for $y$. The graph is a vertical line passing through the $x$-axis at $2$.

What if the equation has $y$ but no $x$? Let’s graph the equation $y = 4$. This time the $y$-value is a constant, so in this equation $y$ does not depend on $x$. Fill in $4$ for all the $y$’s in the table below and choose any values for $x$ — we’ll use $0$, $2$, and $4$.

The graph is a horizontal line passing through the $y$-axis at $4$.

Example. Graph the equation $y = -1$.

The equation $y = -1$ has only one variable, $y$. The value of $y$ is constant. All the ordered pairs in its table have the same $y$-coordinate. The graph is a horizontal line passing through the $y$-axis at $-1$.

The equations for vertical and horizontal lines look very similar to equations like $y = 4x$. What is the difference between the equations $y = 4x$ and $y = 4$?

The equation $y = 4x$ has both $x$ and $y$. The value of $y$ depends on the value of $x$, so the $y$-coordinate changes according to the value of $x$. The equation $y = 4$ has only one variable. The value of $y$ is constant — the $y$-coordinate is always $4$. It does not depend on the value of $x$.

Graphing both equations in the same rectangular coordinate system shows the difference clearly: $y = 4x$ gives a slanted line through the origin, while $y = 4$ gives a horizontal line.

Key terms

graph of a linear equation — the line consisting of all the points that are solutions of the equation $Ax + By = C$; every point on the line is a solution of the equation, and every solution of the equation is a point on the line. plotting points (Point-Plotting Method) — a method for graphing a linear equation by finding several ordered-pair solutions, plotting them, and drawing the line through them. vertical line — the graph of an equation of the form $x = a$; a straight up-and-down line that crosses the $x$-axis at $(a, 0)$. horizontal line — the graph of an equation of the form $y = b$; a straight side-to-side line that crosses the $y$-axis at $(0, b)$.

This section is adapted from Elementary Algebra 2e, Section 4.2: Graph Linear Equations in Two Variables by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the coordinate-plane graphs (points, lines, vertical and horizontal lines, and the paired $y = 4x$ vs. $y = 4$ comparison) as accessible inline SVGs and the solution tables as markdown tables; omitted the Be Prepared quiz, Self Check checklist, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.