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Linear — Definition, Formula & Examples

Linear

Like a line. A description of any graph or data that can be modeled by a linear polynomial.

Key Formula

y=mx+by = mx + b
Where:
  • yy = The output (dependent variable)
  • mm = The slope — the constant rate of change of y with respect to x
  • xx = The input (independent variable)
  • bb = The y-intercept — the value of y when x = 0

Worked Example

Problem: Determine whether the equation y = 3x + 5 is linear, and find the slope and y-intercept.
Step 1: Check the form of the equation. Compare it to the standard linear form y = mx + b.
y=3x+5y = 3x + 5
Step 2: Verify that x appears only to the first power. Here x has exponent 1, so the equation is linear.
Step 3: Identify the slope m by reading the coefficient of x.
m=3m = 3
Step 4: Identify the y-intercept b by reading the constant term.
b=5b = 5
Answer: The equation y = 3x + 5 is linear with slope 3 and y-intercept 5. Its graph is a straight line that crosses the y-axis at (0, 5) and rises 3 units for every 1 unit increase in x.

Another Example

This example tests whether real data is linear by checking for a constant rate of change, rather than starting from a given equation.

Problem: A table shows the following data: when x = 0, y = 4; when x = 2, y = 10; when x = 4, y = 16; when x = 6, y = 22. Determine whether the data is linear and, if so, find the equation.
Step 1: Calculate the rate of change between consecutive data points. From x = 0 to x = 2:
10−42−0=62=3\frac{10 - 4}{2 - 0} = \frac{6}{2} = 3
Step 2: Check the rate of change for the next pair. From x = 2 to x = 4:
16−104−2=62=3\frac{16 - 10}{4 - 2} = \frac{6}{2} = 3
Step 3: Check one more pair. From x = 4 to x = 6:
22−166−4=62=3\frac{22 - 16}{6 - 4} = \frac{6}{2} = 3
Step 4: The rate of change is constant at 3, so the data is linear with slope m = 3. Since y = 4 when x = 0, the y-intercept is b = 4.
y=3x+4y = 3x + 4
Answer: The data is linear. The equation is y = 3x + 4.

Frequently Asked Questions

What makes an equation linear vs. nonlinear?
An equation is linear if every variable appears only to the first power and no two variables are multiplied together. For example, y = 2x + 7 is linear, but y = x² + 1 is nonlinear because x is squared, and y = xy + 3 is nonlinear because x and y are multiplied. The key test: a linear equation always graphs as a straight line.
How do you tell if a graph is linear?
A graph is linear if all its points lie on a single straight line. You can verify this by checking that the slope (rise over run) between any two points on the graph is the same. If the line curves, bends, or changes steepness, the relationship is not linear.
Can a linear equation have two variables?
Yes. The most common linear equation has two variables: y = mx + b. More generally, any equation of the form ax + by = c (where a, b, and c are constants) is a linear equation in two variables. Its graph is always a straight line in the coordinate plane.

Linear vs. Nonlinear

LinearNonlinear
DefinitionRelationship with a constant rate of changeRelationship with a changing rate of change
Highest exponent on variables1 (first power only)2 or higher, or variables in denominators, exponents, etc.
Graph shapeStraight lineCurve (parabola, hyperbola, etc.)
Example equationy = 4x − 2y = x² + 3x − 2
Rate of changeConstant (same slope everywhere)Variable (slope changes from point to point)

Why It Matters

Linearity is one of the most fundamental concepts in algebra and appears in nearly every math course from middle school through college. You use linear models to represent real-world situations with constant rates of change—distance traveled at steady speed, cost per item, temperature conversions, and more. Recognizing whether a relationship is linear or not is the first step in choosing the right equation or model to solve a problem.

Common Mistakes

Mistake: Assuming any equation with x in it is linear.
Correction: Equations like y = x², y = √x, and y = 1/x all contain x but are nonlinear. For an equation to be linear, every variable must appear to exactly the first power with no variable-to-variable products.
Mistake: Confusing a linear equation with a linear expression.
Correction: A linear expression like 3x + 2 has no equals sign—it is just a polynomial of degree 1. A linear equation, such as y = 3x + 2 or 3x + 2 = 11, sets two expressions equal and can be solved or graphed.

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