Standard form: Ax + By = C, with examples

The standard form of a linear equation is Ax + By = C, where A, B and C are integers (whole numbers) and A is not negative. For example, 2x − 3y = 12 is in standard form. Quadratics have a standard form too, ax² + bx + c = 0. In the UK, standard form of a number means something else: writing it as A × 10ⁿ, which the US calls scientific notation.

What does standard form mean?

Standard form means an agreed way of writing something, so it depends on what you are writing. This page is mostly about straight lines. The other meanings each have a short section further down.

Standard form of a linear equation

The x term comes first, then the y term, then the equals sign, with the number on its own on the right. Most US textbooks also ask for three tidy-up rules:

• A, B and C are integers, so no fractions or decimals.
• A is not negative. If it is, multiply the whole equation by −1.
• A, B and C have no common factor other than 1, and A and B are not both zero.

All five of those describe the same line. Standard form just picks one way to write it, so everyone's answer looks the same.

How to convert slope-intercept form to standard form

1. Clear any fractions by multiplying every term by the lowest common denominator.
2. Move the x term to the left so x and y are on the same side.
3. Make A positive by multiplying everything by −1 if you need to.

Example 1: y = ⅔x − 4

Multiply every term by 3 to get 3y = 2x − 12. Subtract 2x from both sides: −2x + 3y = −12. A is negative, so multiply through by −1: 2x − 3y = 12.

Example 2: two different fractions

Write y = (5/2)x − 7/3 in standard form. The denominators are 2 and 3, so multiply every term by 6:

The answer is 15x − 6y = 14. Check for a common factor: 15, 6 and 14 share none, so it is finished.

How to convert standard form to slope-intercept form

Make y the subject. For 3x + 4y = 10, subtract 3x to get 4y = −3x + 10, then divide by 4:

In general, the line Ax + By = C has:

Check with 3x + 4y = 10: the slope is −3/4 and the y-intercept is 10/4 = 5/2, matching the working above.

Graphing a line in standard form

Standard form makes the intercepts quick, and two points are all you need for a line. Put y = 0 to find where it crosses the x-axis, and x = 0 for the y-axis. For 2x − 3y = 12:

• y = 0 gives 2x = 12, so x = 6. The x-intercept is (6, 0).
• x = 0 gives −3y = 12, so y = −4. The y-intercept is (0, −4).

Here x and y are the usual coordinates. Plot the two intercepts and join them. The slope is −2/(−3) = ⅔, so the line rises 2 for every 3 across. Try other lines in the linear graph calculator.

Writing an equation in standard form

Example 3: from a point and a slope

Find the standard form of the line through (1, 5) with slope −3. Point-slope form gives y − 5 = −3(x − 1). Expand: y − 5 = −3x + 3, so y = −3x + 8. Add 3x to both sides: 3x + y = 8. Check with the point: 3(1) + 5 = 8.

Example 4: from two points

Find the standard form of the line through (−1, 5) and (3, −3). The slope is (−3 − 5) ÷ (3 − (−1)) = −8 ÷ 4 = −2. Then y − 5 = −2(x + 1), so y = −2x + 3, and 2x + y = 3. Check the other point: 2(3) + (−3) = 3.

Horizontal and vertical lines

When A = 0, the x term disappears and you get a horizontal line such as y = 4. When B = 0, you get a vertical line such as x = −2. A vertical line has no slope-intercept form at all, because its slope is undefined, but standard form handles it without trouble. That is one reason some teachers prefer it.

Why use standard form?

Word problems

Problems about two things adding up to a total come out in standard form naturally. A school play sells adult tickets for $8 and child tickets for $5, and takes $200. With x adult tickets and y child tickets:

The intercepts mean something here: if only adults came, 200 ÷ 8 = 25 tickets were sold, and if only children came, 200 ÷ 5 = 40. Every whole-number point on the line between them is a possible mix, such as 15 adults and 16 children.

Solving systems by elimination

With two equations in standard form, the x terms and y terms line up in columns, ready to add or subtract. For 2x + y = 3 and x − y = 3, adding the equations cancels y: 3x = 6, so x = 2, and then y = −1. Try one now, then press Mark to check it in the course.

The simultaneous equations calculator solves systems like this step by step, and the Algebra course teaches elimination and substitution from the start.

Parallel and perpendicular lines

Lines with the same A and B but a different C are parallel, because they have the same slope. So the line parallel to 3x + 4y = 10 through (2, 3) is 3x + 4y = 18, since 3(2) + 4(3) = 18. To get a perpendicular line, swap A and B and change one sign: the line perpendicular to 3x + 4y = 10 through (2, 3) is 4x − 3y = −1. More on this in perpendicular lines.

Standard form of a quadratic

A quadratic equation is in standard form when everything is on one side, in order of power, equal to zero: ax² + bx + c = 0, with a ≠ 0. This is the form the quadratic formula needs. Expand any brackets to get there: (x − 3)(x + 5) becomes x² + 2x − 15, and 2(x − 1)² + 3 becomes 2x² − 4x + 5. A few precalculus textbooks use "standard form of a quadratic function" for the vertex form a(x − h)² + k instead, so check which one your course means. The quadratic graph calculator plots any quadratic in standard form.

Standard form of a polynomial or a circle

A polynomial is in standard form when its terms run from the highest power of x to the lowest, so 5 + 3x³ − x becomes 3x³ − x + 5. A circle with centre (center in the US) (h, k) and radius r has the standard form equation (x − h)² + (y − k)² = r², so (x − 2)² + (y + 1)² = 9 is centred on (2, −1) with radius 3.

Standard form of a number

In the UK, standard form means writing a number as A × 10ⁿ, where A is at least 1 and less than 10 and n is an integer, so 45,000 = 4.5 × 10⁴ and 0.00072 = 7.2 × 10⁻⁴. AQA's GCSE Maths specification uses this definition, and in the US the same thing is taught in grade 8 as scientific notation. The scientific notation calculator converts numbers both ways with working.

In US elementary school, standard form of a number just means writing it with digits, such as 3,482, as opposed to expanded form (3,000 + 400 + 80 + 2) or word form.

Common mistakes

• Leaving fractions in. Multiply every term, including C, by the common denominator.
• Leaving A negative. −2x + 3y = −12 is the right line but not standard form. Multiply by −1.
• Changing only some signs. When you multiply by −1, every term changes sign, on both sides.
• Reading the slope as A. In 2x − 3y = 12 the slope is −A/B = ⅔, not 2.
• Forgetting a common factor. 4x − 6y = 24 should be simplified to 2x − 3y = 12.

To practise linear equations among other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 or Geometry, Algebra 2 & SAT Math paper.

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