Standard Form

Q: What do A, B, and C have to be?

In most textbooks $A$, $B$, and $C$ are integers, $A$ and $B$ are not both zero, and $A$ is usually written as a positive number. So $2x - 5y = 15$ is standard form, while $y = 3x - 5$ and $y - 2 = 4(x - 1)$ are not.

Q: How do I find the slope from standard form?

Either solve for $y$ and read the coefficient of $x$, or use the shortcut: the slope of $Ax + By = C$ is $-\dfrac{A}{B}$. For $2x + 3y = 9$ that gives $-\dfrac{2}{3}$.

Standard form is one of the three main ways to write a linear equation: $Ax + By = C$ , where $A$ , $B$ , and $C$ are numbers and the $x$ and $y$ terms sit together on the left side. The equation $3x + 4y = 12$ is in standard form; $y = 2x + 5$ is not.

Standard form hides the slope, but it makes one thing incredibly easy: finding the intercepts. Set $y = 0$ and you get the $x$ -intercept. Set $x = 0$ and you get the $y$ -intercept. Two intercepts is two points, and two points is all you need to graph the line.

Finding the intercepts

The $x$ -intercept is where the line crosses the x-axis. Every point on the x-axis has $y = 0$ , so plug in $y = 0$ and solve for $x$ . In $3x + 4y = 12$ , that gives $3x = 12$ , so $x = 4$ and the $x$ -intercept is $(4, 0)$ .

The $y$ -intercept is where the line crosses the y-axis, where $x = 0$ . Plug in $x = 0$ : $4y = 12$ , so $y = 3$ and the $y$ -intercept is $(0, 3)$ .

Notice the shortcut: setting one variable to zero wipes out its whole term, so each intercept is a one-step division. That is why standard form is the fastest form to graph from.

Graphing with two intercepts

Plot the $x$ -intercept on the x-axis, plot the $y$ -intercept on the y-axis, and draw the line through both. The graph below shows $2x + 3y = 12$ , which has $x$ -intercept $(6, 0)$ and $y$ -intercept $(0, 4)$ .

Converting to slope-intercept form

Sometimes you need the slope, and standard form does not show it directly. To convert $Ax + By = C$ into $y = mx + b$ , isolate $y$ : subtract the $x$ term from both sides, then divide everything by the coefficient of $y$ .

For example, $2x + 3y = 9$ becomes $3y = -2x + 9$ , and dividing by $3$ gives $y = -\dfrac{2}{3}x + 3$ . Now you can read the slope, $-\dfrac{2}{3}$ , straight off the equation. In general the slope of $Ax + By = C$ works out to $-\dfrac{A}{B}$ .

Worked examples

Example 1: find both intercepts

Find the $x$ -intercept and $y$ -intercept of $2x + 3y = 12$ .

For the

x

-intercept, set

y = 0

2x + 3(0) = 12

Solve for

x

2x = 12 \;\Rightarrow\; x = 6

For the

y

-intercept, set

x = 0

2(0) + 3y = 12

Solve for

y

3y = 12 \;\Rightarrow\; y = 4

Answer: $x$ -intercept $(6, 0)$ , $y$ -intercept $(0, 4)$

Example 2: convert to slope-intercept form

Write $4x + 2y = 10$ in slope-intercept form.

Start with the equation

4x + 2y = 10

Subtract

4x

from both sides

2y = -4x + 10

Divide both sides by

2

y = -2x + 5

Answer: $y = -2x + 5$

Example 3: find the slope without fully converting

What is the slope of $5x - 2y = 8$ ?

Move the

x

term

-2y = -5x + 8

Divide both sides by

-2

y = \dfrac{5}{2}x - 4

Read the coefficient of

x

m = \dfrac{5}{2}

Answer: $m = \dfrac{5}{2}$

Try one yourself

Common questions

What do A, B, and C have to be?

In most textbooks $A$ , $B$ , and $C$ are integers, $A$ and $B$ are not both zero, and $A$ is usually written as a positive number. So $2x - 5y = 15$ is standard form, while $y = 3x - 5$ and $y - 2 = 4(x - 1)$ are not.

When is standard form better than slope-intercept form?

When you want intercepts or a fast graph. Slope-intercept form $y = mx + b$ is better when you need the slope or want to compare steepness. They describe the same line — pick the form that makes your task a one-step read.

How do I find the slope from standard form?

Either solve for $y$ and read the coefficient of $x$ , or use the shortcut: the slope of $Ax + By = C$ is $-\dfrac{A}{B}$ . For $2x + 3y = 9$ that gives $-\dfrac{2}{3}$ .

What if an intercept comes out as a fraction?

That is fine — the line just crosses the axis between grid points. For $3x + 4y = 10$ , setting $y = 0$ gives $x = \dfrac{10}{3}$ . Plot it about a third of the way past $3$ and keep going.

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Standard Form

Finding the intercepts

Graphing with two intercepts

Converting to slope-intercept form

Worked examples

Example 1: find both intercepts

Example 2: convert to slope-intercept form

Example 3: find the slope without fully converting

Try one yourself

Common questions

Keep learning

Slope-Intercept Form

Point-Slope Form

Equations of Parallel Lines

The Slope Formula