The standard form of a linear equation is Ax + By = C, where A, B and C are integers and A and B are not both 0. Both variables sit on the left side. It makes the intercepts easy to find: the x-intercept is CA, the y-intercept is CB, and the slope is −AB.
In Spanish
forma estándar de una ecuación lineal
Also called
standard form, Ax + By = C, general form of a line
In standard form, setting one variable to 0 gives an intercept in one step, so two dots and a ruler graph the line.
Formal definition and notation
A linear equation in two variables is in standard form when it is written Ax + By = C with integer coefficients A, B and C, where A and B are not both zero. Most US textbooks also ask that A ≥ 0 and that A, B and C have no common factor other than 1, so each line has exactly one standard form. Every line in the coordinate plane, including vertical lines (B = 0) and horizontal lines (A = 0), can be written this way.
Symbol
Read it as
Meaning
A, B
A, B
the integer coefficients of x and y; some books require A ≥ 0
Answer: The x-intercept is (5, 0) and the y-intercept is (0, 4).
Example 2: Rewrite slope-intercept form
Core
Write y = −34x + 2 in standard form with integer coefficients and A > 0.
Multiply every term by 4 to clear the fraction: 4y = −3x + 8.
Add 3x to both sides: 3x + 4y = 8.
Check: A = 3 is positive, and 3, 4, 8 share no common factor.
Answer:3x + 4y = 8
Example 3: A ticket-sales model
Test-style
A school play charges 8 dollars per adult ticket and 5 dollars per student ticket. It collected 200 dollars. Write an equation in standard form, then find the number of student tickets sold if 15 adult tickets were sold.
Let a be adult tickets and s be student tickets. Money from each type adds to 200: 8a + 5s = 200.
Slope-intercept form y = mx + b shows the slope and y-intercept directly; standard form Ax + By = C keeps integers and shows both intercepts quickly. y = 2x − 3 and 2x − y = 3 are the same line.
Point-slope form y − y1 = m(x − x1) is built from one point and the slope; rearrange it to reach standard form.
Standard Form of a Linear Equationvs.Standard form of a quadratic or a number
“Standard form” also names ax2 + bx + c = 0 for quadratics and, in some countries, scientific notation for numbers. On a linear-equation question it means Ax + By = C.
Quick check: 3 questions
Question 1 of 3
Which equation is in standard form?
Show answer and explanation
Answer: B.2x − y = −7 has both variables on the left, integers, and the constant on the right. y = 2x + 7 is slope-intercept form, 2x = y − 7 splits the variables across the equal sign, and x2 + y = 3 has a fraction.
Question 2 of 3
What is the slope of the line 6x − 2y = 10?
Show answer and explanation
Answer: D. Slope = −AB = −6−2 = 3. −3 forgets that B is negative, 6 is A, and 5 is the y-intercept’s size with the wrong sign (the y-intercept is −5).
Question 3 of 3
Where does 3x + 5y = 30 cross the x-axis?
Show answer and explanation
Answer: B. On the x-axis y = 0, so 3x = 30 and x = 10: the point (10, 0). (0, 6) is the y-intercept, and the other two mix up which coefficient to divide by.
Standard Form of a Linear Equation: frequently asked questions
How do you convert slope-intercept form to standard form?
Move the x term to the left side so both variables are together, then clear fractions and make A positive. For y = 25x − 3: multiply by 5 to get 5y = 2x − 15, subtract 2x to get −2x + 5y = −15, then multiply by −1: 2x − 5y = 15.
Does A have to be positive in standard form?
Most US Algebra 1 textbooks and tests ask for A ≥ 0, integer coefficients and no common factor, so that each line has one standard form. Mathematically, −2x − 3y = −12 is the same line as 2x + 3y = 12, but the second one is the expected answer.
Why use standard form instead of y = mx + b?
Standard form keeps integer coefficients, gives both intercepts in one step each, handles vertical lines such as x = 4, and lines up like terms for the elimination method. It also fits word problems where two kinds of items add to a total, like 8a + 5s = 200.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-CED.A.2, CCSS HSA-REI.D.10
Cite this entry
Effortless Math Team. “Standard Form of a Linear Equation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/standard-form-of-a-linear-equation/
Flashcard
TermStandard Form of a Linear EquationAx + By = C
DefinitionIntegers A, B, C with both variables on the left. Example: 2x + 3y = 12 has intercepts (6, 0) and (0, 4), slope −23.
Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1Standard FormStandard form of a number is the usual way of writing it with digits, such as 4,306, with each digit in its place-value position. It is the short form you get from expanded form (4,000 + 300 + 6) or from word form (four thousand, three hundred six). Zeros hold any empty places.Number System · Grade 2, Grade 3, Grade 4CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Least Common DenominatorThe least common denominator (LCD) of two or more fractions is the smallest number that every denominator divides into evenly. It is the least common multiple of the denominators. For 14 and 56, the LCD is 12, so you rewrite them as 312 and 1012 before you add, subtract or compare.Number System · Grade 5, Grade 6, Grade 7Slope-Intercept FormSlope-intercept form is the linear equation form y = mx + b, where m is the slope and b is the y-intercept. You can read both values directly from the equation without rearranging: in y = 2x + 5, the slope is 2 and the line crosses the y-axis at (0, 5).Algebra · Grade 8, Algebra 1Parallel LinesParallel lines are lines in the same plane that never intersect. They stay the same distance apart and point in the same direction, like the rails of a straight railroad track. On a coordinate plane, parallel lines have the same slope but different y-intercepts. The symbol ∥ means is parallel to.Geometry · Grade 4, Grade 5, Grade 6Elimination MethodThe elimination method solves a system of equations by adding or subtracting them so one variable cancels. Scaling one equation first is often needed to make the coefficients match. After one variable is eliminated, solve for the other variable, then substitute back to find the first.Algebra · Grade 8, Algebra 1Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8Y-InterceptThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Algebra · Grade 8, Algebra 1X-InterceptAn x-intercept is a point where a graph crosses or touches the x-axis. Its y-coordinate is always 0, so it is written (a, 0). To find it, set y = 0 and solve for x. A non-horizontal line has one x-intercept, while a parabola can have two, one or none.Algebra · Grade 8, Algebra 1, Algebra 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1Point-Slope FormPoint-slope form is a way to write the equation of a line from its slope m and one point (x1, y1) on it: y − y1 = m(x − x1). It is the fastest form to use when you know a point and the slope, or two points. Multiplying it out gives slope-intercept form.Algebra · Grade 8, Algebra 1Scientific NotationScientific notation writes a number as a value from 1 to less than 10 times a power of 10, such as 5.4 × 103 for 5,400. The exponent records how far the decimal point moved: positive for large numbers and negative for numbers between 0 and 1, as in 0.0062 = 6.2 × 10−3.Number System · Grade 8, Algebra 1System of EquationsA system of equations is a set of two or more equations that use the same variables and are solved together. Its solution is the values that make every equation true at the same time. For two linear equations, that is the point where the two lines cross. A system can have one solution, none, or infinitely many.Algebra · Grade 8, Algebra 1GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7