How to Graph Linear Equations

Quick answer

A graph of a linear equation is the picture of every solution it has. To draw one, either plot the y-intercept and step by the slope, find the x- and y-intercepts and join them, or build a small table of values. All three give the same line, because a line is determined once you have two points on it.

What you'll learn

Graph a line from slope-intercept form, from intercepts, or from a table
Explain why every point on the line is a solution of the equation
Graph horizontal and vertical lines correctly

What a graph actually is

A graph is not a decoration attached to an equation. It is a picture of every solution.

Take $y = 2x + 1$ . The pair $(3, 7)$ is a solution, because $2(3) + 1 = 7$ . So is $(0, 1)$ , and $(-2, -3)$ , and infinitely many others. Plot them all and they fall into a straight line.

That gives the two facts everything else rests on:

Every point on the line is a solution of the equation. Every solution of the equation is a point on the line.

So “graph the equation” means “show me all the solutions at once”, and “is $(5, 11)$ on the line?” is answered by substituting rather than by squinting at a picture.

Why two points are enough

Through any two distinct points there is exactly one straight line. That is why every method below stops after finding two or three points: once you have them, the line is already decided.

Plot a third point anyway. It costs one substitution and catches arithmetic errors, because three correct points always lie in a straight row.

Method 1 — From slope-intercept form

Best when the equation already looks like $y = mx + b$ .

Plot $b$ on the $y$ -axis.
Step by the slope as $\tfrac{\text{rise}}{\text{run}}$ to a second point.
Repeat once more, then draw the line.

Method 2 — From the intercepts

Best when the equation looks like $Ax + By = C$ , because rearranging is more work than substituting twice.

The $x$ -intercept is where the line crosses the horizontal axis, so $y = 0$ there. The $y$ -intercept is where it crosses the vertical axis, so $x = 0$ there.

Set one variable to zero, solve for the other, and repeat.

Method 3 — From a table

Always works, including for equations that are neither form. Choose a few convenient $x$ values, compute each $y$ , and plot the pairs.

Horizontal and vertical lines

These trip people up because one variable is missing.

Equation	Graph	Why
$y = 3$	horizontal line through $(0,3)$	$y$ is always $3$ , whatever $x$ is
$x = 4$	vertical line through $(4,0)$	$x$ is always $4$ , whatever $y$ is

A memory aid that survives pressure: the equation names the axis it crosses. $x = 4$ crosses the $x$ -axis at $4$ and runs vertically; $y = 3$ crosses the $y$ -axis at $3$ and runs horizontally.

Common mistakes

Practice problems

Is $(2, 9)$ on the line $y = 3x + 3$ ?

Hint

Substitute the $x$ -coordinate and compare with the $y$ -coordinate.

Answer

Yes.

Full solution

$y = 3(2) + 3 = 9$ , which matches, so the point is on the line.
Graph $y = x - 2$ by finding three points.

Answer

$(0, -2)$ , $(1, -1)$ , $(3, 1)$

Full solution

The intercept is $(0, -2)$ . The slope is $1 = \tfrac{1}{1}$ , so step up $1$ across $1$ to $(1, -1)$ , then again to $(2, 0)$ and $(3, 1)$ .

All lie in a straight row, confirming the arithmetic.
Find both intercepts of $2x + 5y = 10$ .

Hint

Set one variable to zero at a time.

Answer

$x$ -intercept $(5, 0)$ , $y$ -intercept $(0, 2)$

Full solution

Set $y = 0$ : $2x = 10$ , so $x = 5$ , giving $(5, 0)$ .

Set $x = 0$ : $5y = 10$ , so $y = 2$ , giving $(0, 2)$ .
Find both intercepts of $4x - 3y = 12$ .

Answer

$x$ -intercept $(3, 0)$ , $y$ -intercept $(0, -4)$

Full solution

Set $y = 0$ : $4x = 12$ , so $x = 3$ .

Set $x = 0$ : $-3y = 12$ , so $y = -4$ .
What does the graph of $y = -1$ look like?

Answer

A horizontal line one unit below the $x$ -axis.

Full solution

$y$ is $-1$ for every value of $x$ , so the solutions are $(-3, -1)$ , $(0, -1)$ , $(5, -1)$ and so on — a horizontal line through $(0, -1)$ .
What does the graph of $x = -2$ look like?

Answer

A vertical line two units left of the $y$ -axis.

Full solution

Only $x$ is constrained, so $y$ can be anything: $(-2, 0)$ , $(-2, 4)$ , $(-2, -3)$ . These stack into a vertical line through $(-2, 0)$ .
Make a table of three points for $y = \tfrac{1}{2}x + 1$ , choosing $x$ values that avoid fractions.

Hint

Halving is clean when $x$ is even.

Answer

$(0, 1)$ , $(2, 2)$ , $(4, 3)$

Full solution

Choosing even $x$ values keeps $\tfrac{1}{2}x$ whole:

$x = 0$ gives $y = 1$ ; $x = 2$ gives $y = 2$ ; $x = 4$ gives $y = 3$ .

Each step of $2$ across raises $y$ by $1$ , matching a slope of $\tfrac{1}{2}$ .
A taxi charges $3 to start plus $2 per mile. Graph the cost $y$ against miles $x$ , and explain why only part of the line makes sense.

Hint

Write the equation first. Then ask which $x$ values are possible in reality.

Answer

$y = 2x + 3$ , but only the part with $x \ge 0$ is meaningful.

Full solution

The flat fee is the intercept and the per-mile rate is the slope: $y = 2x + 3$ .

Points: $(0, 3)$ , $(1, 5)$ , $(3, 9)$ .

The equation has solutions for negative $x$ , such as $(-2, -1)$ , but a journey cannot be $-2$ miles long. The mathematical line continues in both directions; only the portion with $x \ge 0$ describes the real situation. Restricting a graph to the values that make sense is called limiting it to a sensible domain.

Frequently asked questions

How many points do I need?

Two are enough to determine a line, but plot three. The third is a free error check: if it does not land on the straight line through the other two, one of your points is wrong.

Which method should I use?

If the equation is already y = mx + b, stepping from the intercept is fastest. If it is in the form Ax + By = C, finding both intercepts is usually less work than rearranging. A table always works and is worth falling back on.

Why is x = 4 a vertical line rather than a point?

The equation puts no condition on y, so every point whose x-coordinate is 4 satisfies it — (4, 0), (4, 1), (4, -7) and so on. Those points stack into a vertical line.

What to learn next

x-Intercepts and y-Intercepts: Finding Where a Line CrossesFind the x- and y-intercepts of any equation by setting the other variable to zero, why that works, and what intercepts mean in a real situation.

Key terms in this lesson

Linear equation: A linear equation is one whose graph is a straight line. Every variable appears to the first power only, so there are no squares, roots, or variables in a denominator.
Slope: Slope measures how steep a line is: the change in y divided by the change in x. A slope of 2 means the line climbs 2 units for every 1 unit across.
y-intercept: The y-intercept is where a graph crosses the vertical axis — the value of y when x is 0. In y = 2x + 3 the y-intercept is 3, so the line passes through (0, 3).

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSA.REI.D.10Reasoning with Equations and InequalitiesUnderstand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
CCSS.MATH.CONTENT.HSF.IF.C.7aInterpreting FunctionsGraph linear and quadratic functions and show intercepts, maxima, and minima.
CCSS.MATH.CONTENT.8.F.A.3FunctionsInterpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.