Standard Form of a Linear Equation: Ax + By = C

Quick answer

Standard form writes a line as Ax + By = C, with A, B and C usually integers and A not negative. It is the fastest form for finding both intercepts and the only one that can express a vertical line. Its slope is -A/B, and dividing through by B converts it to slope-intercept form.

What you'll learn

Recognise and write a linear equation in standard form
Convert between standard form and slope-intercept form
Explain why standard form can describe lines that y = mx + b cannot

The form

Ax + By = C

Both variables sit on the left and the constant on the right. By convention $A$ , $B$ and $C$ are integers and $A$ is not negative, though the equation still describes the same line if they are not.

$3x + 2y = 12$ is in standard form. $y = 2x + 3$ is not, because the variables are on opposite sides.

What each form is good at

	Slope-intercept $y = mx + b$	Standard $Ax + By = C$
Read the slope	immediately	as $-\tfrac{A}{B}$
Read the $y$ -intercept	immediately	one substitution
Find the $x$ -intercept	one substitution	one substitution
Graph quickly	plot $b$ , step by $m$	plot both intercepts
Describe a vertical line	cannot	$x = 4$ is $1x + 0y = 4$

Neither form is better in general. They answer different questions, and converting between them is a rearrangement you already know from literal equations.

Why standard form can do something slope-intercept cannot

This is the reason the form survives rather than being replaced by the friendlier one.

A vertical line has an undefined slope. Slope-intercept form has an $m$ in it, and there is nothing to put there — no number represents “undefined”. So $x = 4$ has no slope-intercept form at all.

Standard form has no $m$ . It needs only coefficients:

x = 4 \quad\text{is}\quad 1x + 0y = 4

Setting $B = 0$ removes $y$ from the equation entirely, which is exactly what a vertical line means: $x$ is pinned and $y$ is free. Slope-intercept form cannot express that, because it is built to give $y$ in terms of $x$ , and here $y$ depends on nothing.

Where the slope $-\tfrac{A}{B}$ comes from

Solve the general form for $y$ once, and the shortcut is yours forever:

Ax + By = C

By = -Ax + C

y = -\frac{A}{B}x + \frac{C}{B}

Comparing with $y = mx + b$ : the slope is $-\tfrac{A}{B}$ and the $y$ -intercept is $\tfrac{C}{B}$ . This is the same move as deriving the quadratic formula — do the algebra once with letters so you never repeat it with numbers.

Worked examples

Common mistakes

Practice problems

Write $2x + y = 8$ in slope-intercept form.

Hint

Solve for $y$ .

Answer

$y = -2x + 8$

Full solution

Subtract $2x$ from both sides: $y = -2x + 8$ , so $m = -2$ and $b = 8$ .

Shortcut check: $-\tfrac{A}{B} = -\tfrac{2}{1} = -2$ ✓
Write $6x + 3y = 9$ in slope-intercept form.

Answer

$y = -2x + 3$

Full solution

Subtract $6x$ : $3y = -6x + 9$ . Divide every term by $3$ : $y = -2x + 3$ .
Write $y = 4x + 1$ in standard form with $A$ positive.

Answer

$4x - y = -1$

Full solution

Move the $x$ -term: $-4x + y = 1$ . Multiply by $-1$ : $4x - y = -1$ .

Check at $x = 0$ : $-y = -1$ gives $y = 1$ ✓
What is the slope of $5x - 2y = 10$ ?

Hint

Use $-\tfrac{A}{B}$ , minding the sign of $B$ .

Answer

$m = \tfrac{5}{2}$

Full solution

$A = 5$ and $B = -2$ , so $m = -\tfrac{5}{-2} = \tfrac{5}{2}$ .

Confirming by rearranging: $-2y = -5x + 10$ , so $y = \tfrac{5}{2}x - 5$ ✓
Find both intercepts of $7x + 2y = 14$ without converting the equation.

Answer

$(2, 0)$ and $(0, 7)$

Full solution

Set $y = 0$ : $7x = 14$ , so $x = 2$ .

Set $x = 0$ : $2y = 14$ , so $y = 7$ .
Write $x = 9$ in standard form.

Answer

$1x + 0y = 9$

Full solution

Taking $B = 0$ removes $y$ from the equation, which is what a vertical line requires. This line has no slope-intercept form.
Write $y = \tfrac{1}{4}x - \tfrac{3}{2}$ in standard form with integer coefficients.

Hint

Multiply everything by the least common denominator first.

Answer

$x - 4y = 6$

Full solution

Multiply every term by $4$ : $4y = x - 6$ .

Move the $x$ -term: $-x + 4y = -6$ . Multiply by $-1$ : $x - 4y = 6$ .

Check at $x = 0$ : $-4y = 6$ gives $y = -\tfrac{3}{2}$ ✓
Are $2x + 3y = 6$ and $4x + 6y = 18$ parallel, perpendicular, the same line, or none of these?

Hint

Compare the slopes with $-\tfrac{A}{B}$ , then check whether one equation is a multiple of the other.

Answer

Parallel.

Full solution

First slope: $-\tfrac{2}{3}$ . Second slope: $-\tfrac{4}{6} = -\tfrac{2}{3}$ . Equal slopes.

They are not the same line: doubling the first gives $4x + 6y = 12$ , and $12 \neq 18$ . The coefficients scale but the constant does not, so the lines are parallel rather than identical.

Frequently asked questions

Why bother with standard form when slope-intercept is easier to read?

Two reasons. Both intercepts fall out in one substitution each, which makes graphing quick. And standard form can describe a vertical line, which y = mx + b cannot, because a vertical line has no slope to put in place of m.

What is the slope in standard form?

It is -A/B. For 3x + 4y = 12 the slope is -3/4. You can derive it any time by solving for y, but the shortcut saves rearranging when all you want is to compare two lines.

Do A, B and C have to be integers?

Not mathematically, but it is the usual convention, along with keeping A positive. Multiplying through by a denominator clears fractions without changing the line.

What to learn next

Function Notation: What f(x) Actually MeansUnderstand f(x) as a name and an input rather than multiplication, how to evaluate it, and how to read statements like f(3) = 7 in context.

Standards alignment

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

CCSS.MATH.CONTENT.HSA.CED.A.2Creating EquationsCreate equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
CCSS.MATH.CONTENT.HSF.IF.C.7aInterpreting FunctionsGraph linear and quadratic functions and show intercepts, maxima, and minima.

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