Standard Form of a Linear Equation

The standard form of a linear equation is $Ax + By = C$ , where $A$ , $B$ and $C$ are integers and $A$ is not negative. An example is $2x + 3y = 12$ . The $x$ – and $y$ -terms are on the left, and the number is on the right. This form makes intercepts easy to find.

Before this lesson: You should know slope-intercept form, point-slope form, and how to find x- and y-intercepts.

Extra practice: The Algebra 1 Unit 4 Practice Pack (Part 2) has printable problems, answer keys and a cheat sheet for this unit.

What Is Standard Form?

The same line can be written in more than one way. For example, $y = 2x - 3$ and $2x - y = 3$ are the same line. Every point that works in one works in the other.

Standard form puts both variable terms on one side and the plain number on the other side. It is a tidy, agreed-on way to write a line. That makes answers easy to compare.

The rules for standard form

Ax + By = C

$A$ , $B$ and $C$ are integers. That means no fractions and no decimals.
$A$ is not negative. If it is, multiply every term by $-1$ .
$A$ and $B$ are not both 0.
Many books also ask that $A$ , $B$ and $C$ share no common factor. So $4x + 6y = 12$ becomes $2x + 3y = 6$ .

How to Write an Equation in Standard Form

Move the $x$ -term and the $y$ -term to the left side. Move the number to the right side.
If there are fractions, multiply every term by the least common denominator.
If $A$ is negative, multiply every term by $-1$ .
Divide out any common factor, if your class asks for it.

To go the other way, solve for $y$ . From $Ax + By = C$ , you get $y = -\tfrac{A}{B}x + \tfrac{C}{B}$ . So the slope is $m = -\tfrac{A}{B}$ .

Worked Examples

Example 1: Write $y = 3x - 2$ in standard form

Write $y = 3x - 2$ in standard form.

-3x + y = -2

Step 1. Subtract $3x$ from both sides to get the x-term on the left (see two-step equations).

3x - y = 2

Step 2. The A-value is −3, which is negative. Multiply every term by −1 to make it positive.

Answer: $3x - y = 2$ . Here $A = 3$ , $B = -1$ and $C = 2$ . $B$ may be negative. Only $A$ must not be.

Example 2: Clear a fraction

Write $y = \tfrac{2}{3}x - 4$ in standard form.

-\tfrac{2}{3}x + y = -4

Step 1. Subtract $\tfrac{2}{3}x$ from both sides.

3\left(-\tfrac{2}{3}x\right) + 3(y) = 3(-4)

Step 2. Multiply every term by 3 to clear the fraction (see multiplying and dividing fractions).

-2x + 3y = -12

Step 3. Simplify each product.

2x - 3y = 12

Step 4. A is negative, so multiply every term by −1.

Answer: $2x - 3y = 12$ .

Check: Use the y-intercept $(0, -4)$ : $2(0) - 3(-4) = 0 + 12 = 12$ . It works.

Example 3: From point-slope form

Write $y - 5 = -\tfrac{1}{2}(x + 4)$ in standard form.

2(y - 5) = -(x + 4)

Step 1. Multiply both sides by 2. That clears the fraction right away.

2y - 10 = -x - 4

Step 2. Use the distributive property on both sides.

x + 2y - 10 = -4

Step 3. Add $x$ to both sides.

x + 2y = 6

Step 4. Add 10 to both sides.

Answer: $x + 2y = 6$ .

Check: Use the point $(-4, 5)$ : $-4 + 2(5) = -4 + 10 = 6$ . It works.

Example 4: Graph $3x + 4y = 12$

Graph $3x + 4y = 12$ . Standard form makes the intercepts quick to find.

3x = 12

Step 1. Let $y = 0$ and simplify.

x = 4

Step 2. Divide both sides by 3. The x-intercept is $(4, 0)$ .

4y = 12

Step 3. Let $x = 0$ and simplify.

y = 3

Step 4. Divide both sides by 4. The y-intercept is $(0, 3)$ .

Step 5. Plot both points and draw the line (see x and y intercepts).

m = -\tfrac{3}{4}

Step 6. Find the slope with $m = -\tfrac{A}{B}$ .

The line 3x + 4y = 12 on a grid with x from −2 to 6 and y from −2 to 5. It crosses the x-axis at (4, 0) and the y-axis at (0, 3), both marked with dots. Beside the graph, the same line is written in two forms: standard form 3x + 4y = 12 and slope-intercept form y = −3/4 x + 3. — In standard form, setting x = 0 or y = 0 gives the intercepts fast: (4, 0) and (0, 3).

Answer: The line through $(4, 0)$ and $(0, 3)$ . In slope-intercept form, it is $y = -\tfrac{3}{4}x + 3$ .

Example 5: Word problem with movie tickets

Adult tickets cost $9 and child tickets cost $6. A group spends exactly $54. Write an equation in standard form. Then find every whole-number answer.

9a + 6c = 54

Step 1. Let $a$ be the number of adult tickets and $c$ the number of child tickets. The cost of adult tickets plus the cost of child tickets is 54.

3a + 2c = 18

Step 2. This is already in standard form. Divide every term by 3 to make the numbers smaller.

Step 3. Find the intercepts. If $c = 0$ , then $3a = 18$ , so $a = 6$ . If $a = 0$ , then $2c = 18$ , so $c = 9$ .

3(2) + 2c = 18

Step 4. Try $a = 2$ . Substitute it.

6 + 2c = 18

Step 5. Multiply.

2c = 12

Step 6. Subtract 6 from both sides.

c = 6

Step 7. Divide both sides by 2.

Step 8. Try $a = 4$ the same way: $12 + 2c = 18$ , so $2c = 6$ and $c = 3$ . Odd values of $a$ give fractions of a ticket, so skip them.

Graph of 9a + 6c = 54, with adult tickets a on the horizontal axis from 0 to 8 and child tickets c on the vertical axis from 0 to 10. The line goes from (0, 9) to (6, 0). The whole-number solutions (0, 9), (2, 6), (4, 3) and (6, 0) are marked with dots and listed beside the graph. — Each dot is a way to spend exactly $54: the intercepts are all adults or all children.

Answer: $9a + 6c = 54$ . The group could buy 0 adult and 9 child tickets, 2 and 6, 4 and 3, or 6 and 0.

Common Mistakes

Mistake	Example of the error	Fix
Sign error when moving a term	$y = 3x - 2$ becomes $3x + y = -2$	Subtract $3x$ from both sides: $-3x + y = -2$ , then $3x - y = 2$ .
Leaving A negative	Stopping at $-2x + 3y = -12$	Multiply every term by $-1$ : $2x - 3y = 12$ .
Clearing a fraction in only one term	$-\tfrac{2}{3}x + y = -4$ becomes $-2x + y = -4$	Multiply every term by 3: $-2x + 3y = -12$ .
Reading the slope as A over B	For $3x + 4y = 12$ , saying $m = \tfrac{3}{4}$	The slope is $-\tfrac{A}{B} = -\tfrac{3}{4}$ .

Changing Between Forms

Start with	To get standard form	Example
Slope-intercept, $y = mx + b$	Move $mx$ left, clear fractions, make $A$ positive	$y = -2x + 5$ becomes $2x + y = 5$
Point-slope, $y - y_1 = m(x - x_1)$	Distribute, then move terms	$y - 1 = 3(x - 2)$ becomes $3x - y = 5$
Standard, $Ax + By = C$	To get slope-intercept, solve for $y$	$5x + 2y = 8$ becomes $y = -\tfrac{5}{2}x + 4$

Standard form can also show lines that slope-intercept form can’t. The vertical line $x = 3$ is $1x + 0y = 3$ . Read more in horizontal and vertical lines.

Later, you will line up two equations in standard form to solve a system. That is the elimination method.

Practice Problems

Write $y = 4x + 7$ in standard form.
Write $y = -2x + 5$ in standard form.
Write $y = \tfrac{1}{2}x - 3$ in standard form.
Write $y = -\tfrac{3}{5}x + 2$ in standard form.
Write $y - 1 = 3(x - 2)$ in standard form.
Write $5x + 2y = 8$ in slope-intercept form.
Find the x-intercept, the y-intercept and the slope of $6x - 3y = 18$ .
Is $-4x + 2y = 9$ in standard form? If not, fix it.
Write the equation of the line through $(1, 4)$ and $(3, 0)$ in standard form.
Granola bars cost $2 and juice boxes cost $3. Write an equation in standard form for spending exactly $24 on $g$ bars and $j$ juices. Find both intercepts and one more whole-number solution.

Show answers

-4x + y = 7

Subtract 4x from both sides.

4x - y = -7

A is negative, so multiply every term by −1.

2x + y = 5

Add 2x to both sides.

A = 2 is already positive, so this is standard form.

-\tfrac{1}{2}x + y = -3

Subtract the x-term from both sides.

x - 2y = 6

Multiply every term by −2. This clears the fraction and makes A positive.

\tfrac{3}{5}x + y = 2

Add the x-term to both sides.

3x + 5y = 10

Multiply every term by 5.

y - 1 = 3x - 6

Distribute the 3.

-3x + y - 1 = -6

Subtract 3x from both sides.

-3x + y = -5

Add 1 to both sides.

3x - y = 5

Multiply every term by −1.

2y = -5x + 8

Subtract 5x from both sides.

y = -\tfrac{5}{2}x + 4

Divide every term by 2.

6x = 18, \text{ so } x = 3

Let y = 0.

-3y = 18, \text{ so } y = -6

Let x = 0.

m = -\tfrac{A}{B} = -\tfrac{6}{-3} = 2

Find the slope.

The intercepts are $(3, 0)$ and $(0, -6)$ , and the slope is 2.

Check A. It is −4, which is negative, so this is not standard form

4x - 2y = -9

Multiply every term by −1.

m = \tfrac{0 - 4}{3 - 1} = \tfrac{-4}{2} = -2

Find the slope.

y - 4 = -2(x - 1)

Use point-slope form with (1, 4)

y - 4 = -2x + 2

Distribute the −2.

y = -2x + 6

Add 4 to both sides.

2x + y = 6

Add 2x to both sides.

10.

2g + 3j = 24

Write the cost equation.

2g = 24, \text{ so } g = 12

Let j = 0.

3j = 24, \text{ so } j = 8

Let g = 0.

Check another pair, 6 bars and 4 juices: $2(6) + 3(4) = 12 + 12 = 24$
The intercepts are $(12, 0)$ , 12 bars and no juice, and $(0, 8)$ , 8 juices and no bars. Another solution is 6 bars and 4 juices.

Want more practice? The Algebra 1 Unit 4 Practice Pack (Part 2) has 38 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Standard Form FAQ

What is the standard form of a linear equation?

It is $Ax + By = C$ , where $A$ , $B$ and $C$ are integers and $A$ is not negative. For example, $4x - 5y = 20$ .

Can A be negative in standard form?

Most textbooks say no. If $A$ is negative, multiply every term by $-1$ . The line stays the same.

How do you find the slope from standard form?

Use $m = -\tfrac{A}{B}$ . For $2x + 5y = 10$ , the slope is $-\tfrac{2}{5}$ . You can also solve for $y$ and read the slope.

Why use standard form?

It makes the intercepts easy to find. It also keeps whole numbers, which helps in word problems and when solving systems of equations.

Summary

Standard form is $Ax + By = C$ , with integer $A$ , $B$ and $C$ , and $A$ not negative.
Move the variable terms left and the number right. Clear fractions by multiplying every term.
If $A$ is negative, multiply every term by $-1$ .
The slope is $-\tfrac{A}{B}$ . The intercepts come from setting $x$ or $y$ to 0.
Many word problems with two kinds of items fit standard form.

Previous lesson: Point-Slope Form

Next lesson: Parallel and Perpendicular Lines

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