A 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.
In Spanish
ecuación lineal
Also called
first-degree equation, equation of a line, linear equations
All the solutions of 2x + 3y = 12 line up on one straight line, which is what makes the equation linear. Squaring x bends the graph into a curve.
Formal definition and notation
A linear equation in one variable can be written ax + b = 0 with a ≠ 0; it has exactly one solution, x = −ba. (Equations that simplify to a = a or a = b have infinitely many or no solutions.) A linear equation in two variables can be written in standard formAx + By = C, with A and B not both 0; its solutions are the ordered pairs(x, y) that make it true, and they form a straight line in the coordinate plane. Each term is a constant or a constant times a single variable to the first power, which is why it is also called a first-degree equation.
A vertical line such as x = 4 is a linear equation but not a function.
Solving a one-variable linear equation
Solving a one-variable linear equation
ax + b = c ⇒ x = c − ba, for a ≠ 0
simplify
distribute and combine like terms on each side
collect
move the variable terms to one side and the constants to the other
isolate
divide by the coefficient of the variable
check
substitute the answer into the original equation
Undo addition and subtraction before multiplication and division: the reverse of the order of operations.
Try it: drag a line, read its equation
Interactive
Drag point A or B, or select a point and use the arrow keys. The readout writes the equation of the line through them. Try the presets: a horizontal line has an equation like y = 2, and a vertical line one like x = 2.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Check: 5(9 − 2) = 35 and 3(9) + 8 = 35. Both sides match.
Answer:x = 9
Example 3: Write and use a linear equation
Test-style
A taxi charges 3 dollars to start plus 2 dollars per mile. Write a linear equation for the cost C of an m-mile ride. How long is a ride that costs 27 dollars?
The cost starts at 3 and rises 2 for every mile: C = 2m + 3. The slope is 2 dollars per mile and the intercept is 3 dollars.
Set C = 27: 2m + 3 = 27.
Subtract 3: 2m = 24. Divide by 2: m = 12.
Check: 2(12) + 3 = 27.
Answer:C = 2m + 3; a 27-dollar ride is 12 miles long.
Common mistakes
✗ Mistakey = x2 + 3 is linear, because it looks like y = mx + b.
✓ InsteadIt is not linear: the x2 term makes its graph a curve (a parabola).
An equation with a squared variable is not linear. In a linear equation every variable has exponent 1.
✗ Mistake3x + 5 = 20: divide by 3 first, x + 5 = 203.
✓ InsteadSubtract 5 first: 3x = 15, then divide: x = 5.
Dividing only the 3x term and the right side breaks the balance. Undo addition and subtraction first, or divide every term by 3.
✗ Mistake2(x + 4) = 18 becomes 2x + 4 = 18.
✓ Instead2(x + 4) = 2x + 8, so 2x + 8 = 18 and x = 5.
The 2 multiplies every term in the parentheses, including the 4.
✗ Mistakexy = 6 is linear, because both variables have exponent 1.
✓ Insteadxy = 6 is not linear; its graph is a curve through (1, 6), (2, 3) and (3, 2).
A term with two variables multiplied together is of degree 2. Linear terms have at most one variable each.
A linear function is a rule f(x) = mx + b that gives one output per input. Every non-vertical linear equation in x and y defines one, but x = 4 is a linear equation that is not a function.
A quadratic equation has a squared variable, as in x2 − 5x + 6 = 0, and can have two solutions. Its graph is a parabola, not a line.
Quick check: 3 questions
Question 1 of 3
Which equation is linear?
Show answer and explanation
Answer: B. In 2x − 5y = 10 each variable appears to the first power, alone in its term, so the graph is a line. Choice A has x2, choice C has x in a denominator, and choice D multiplies two variables.
Question 2 of 3
Solve 6x + 4 = 2x − 12.
Show answer and explanation
Answer: C. Subtract 2x: 4x + 4 = −12. Subtract 4: 4x = −16, so x = −4. The other choices come from sign slips, such as adding 4 instead of subtracting it.
Question 3 of 3
Which is the equation of the line with slope 2 that crosses the y-axis at −3?
Show answer and explanation
Answer: D. In y = mx + b, m is the slope and b the y-intercept: y = 2x − 3. Choice A swaps slope and intercept, choice B drops the negative sign, and choice C swaps them and the sign.
Linear Equation: frequently asked questions
How can you tell if an equation is linear?
Simplify it and look at each term. It is linear if every variable has exponent 1, no term multiplies two variables, and no variable sits in a denominator, under a root or in an exponent. 3x − 4y = 7 and y = x2 + 1 are linear; y = x2, y = √x and y = 2x are not.
How many solutions does a linear equation have?
A linear equation in one variable usually has exactly one solution, like x = 4. It has none if it simplifies to something false, such as 3 = 5, and infinitely many if it simplifies to something always true, such as 3 = 3. A linear equation in two variables has infinitely many solutions: every point on its line.
What are the forms of a linear equation?
Slope-intercept form, y = mx + b, shows the slope m and y-intercept b. Standard form, Ax + By = C, makes the intercepts easy to find. Point-slope form, y − y1 = m(x − x1), uses one point and the slope. All three describe the same kind of straight line and can be rewritten into each other.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.EE.C.7, CCSS 8.F.A.3
Cite this entry
Effortless Math Team. “Linear Equation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/linear-equation/
Flashcard
TermLinear Equationy = mx + b, 3x + 2 = 14
DefinitionAn equation whose graph is a straight line; each variable has exponent 1. 3x + 2 = 14 gives x = 4.
SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6Standard 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 4Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7Coordinate 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 7PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.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 1SlopeSlope 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 1Y-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 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 1CoefficientA 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 8Rate of ChangeA rate of change tells how much an output changes for each 1-unit change in the input: the change in y divided by the change in x. If a phone plan costs 15 dollars more for every extra hour, its rate of change is 15 dollars per hour. For a straight line, the rate of change equals the slope.Functions · Grade 8, Algebra 1, Algebra 2FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3SideA 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 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8GraphA 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 7System 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 1InequalityAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Algebra · Grade 6, Grade 7, Grade 8Standard Form of a Linear EquationThe 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.Algebra · Grade 8, Algebra 1