Standard Form of a Linear Equation

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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.

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

  1. Move the x-term and the y-term to the left side. Move the number to the right side.
  2. If there are fractions, multiply every term by the least common denominator.
  3. If A is negative, multiply every term by -1.
  4. 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

Subtract 3x from both sides. Then fix the negative A.

\begin{aligned} y &= 3x - 2 \\ -3x + y &= -2 \\ 3x - y &= 2 \end{aligned}

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.

First move the x-term. Then multiply every term by 3 to clear the fraction.

\begin{aligned} -\tfrac{2}{3}x + y &= -4 \\ 3\left(-\tfrac{2}{3}x\right) + 3(y) &= 3(-4) \\ -2x + 3y &= -12 \\ 2x - 3y &= 12 \end{aligned}

Answer: 2x - 3y = 12. Check with the y-intercept (0, -4): 2(0) - 3(-4) = 12. It works.

Example 3: From point-slope form

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

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

\begin{aligned} 2(y - 5) &= -(x + 4) \\ 2y - 10 &= -x - 4 \\ x + 2y &= 6 \end{aligned}

Answer: x + 2y = 6. Check the point (-4, 5): -4 + 2(5) = 6. It works.

Example 4: Graph 3x + 4y = 12

Standard form makes the intercepts quick to find.

  • Let y = 0: 3x = 12, so x = 4. The x-intercept is (4, 0).
  • Let x = 0: 4y = 12, so y = 3. The y-intercept is (0, 3).

Plot both points and draw the line. To find the slope, use m = -\tfrac{A}{B} = -\tfrac{3}{4}. In slope-intercept form, the line is y = -\tfrac{3}{4}x + 3.

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).

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.

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.

9a + 6c = 54

This is already in standard form. You can divide by 3 to get 3a + 2c = 18. Now find the intercepts. If c = 0, then a = 6. If a = 0, then c = 9.

Tickets come in whole numbers. Try a = 0, 2, 4 and 6. Each gives a whole c.

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

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

Changing Between Forms

Start withTo get standard formExample
Slope-intercept, y = mx + bMove mx left, clear fractions, make A positivey = -2x + 5 becomes 2x + y = 5
Point-slope, y - y_1 = m(x - x_1)Distribute, then move termsy - 1 = 3(x - 2) becomes 3x - y = 5
Standard, Ax + By = CTo get slope-intercept, solve for y5x + 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

  1. Write y = 4x + 7 in standard form.
  2. Write y = -2x + 5 in standard form.
  3. Write y = \tfrac{1}{2}x - 3 in standard form.
  4. Write y = -\tfrac{3}{5}x + 2 in standard form.
  5. Write y - 1 = 3(x - 2) in standard form.
  6. Write 5x + 2y = 8 in slope-intercept form.
  7. Find the x-intercept, the y-intercept and the slope of 6x - 3y = 18.
  8. Is -4x + 2y = 9 in standard form? If not, fix it.
  9. Write the equation of the line through (1, 4) and (3, 0) in standard form.
  10. 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
  1. -4x + y = 7, so 4x - y = -7.
  2. 2x + y = 5
  3. -\tfrac{1}{2}x + y = -3. Multiply by -2: x - 2y = 6.
  4. \tfrac{3}{5}x + y = 2. Multiply by 5: 3x + 5y = 10.
  5. y - 1 = 3x - 6, so -3x + y = -5. Multiply by -1: 3x - y = 5.
  6. 2y = -5x + 8, so y = -\tfrac{5}{2}x + 4.
  7. x-intercept: 6x = 18, so (3, 0). y-intercept: -3y = 18, so (0, -6). Slope: -\tfrac{6}{-3} = 2.
  8. No, because A = -4 is negative. Multiply by -1: 4x - 2y = -9.
  9. m = \tfrac{0 - 4}{3 - 1} = -2. Then y - 4 = -2(x - 1), so y = -2x + 6. Standard form: 2x + y = 6.
  10. 2g + 3j = 24. Intercepts: (12, 0), which is 12 bars and no juice, and (0, 8), which is 8 juices and no bars. Another solution: 6 bars and 4 juices, since 2(6) + 3(4) = 24.

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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