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Linear Equation — Definition, Formula & Examples

Linear Equation

An equation that can be written in the form "linear polynomial = linear polynomial" or "linear polynomial = constant".

The following are examples of linear equations: 2x – 3 = 5, 4a + 9 = 8 – 9a, and 2x + 5y = 1.

 

 

See also

Linear polynomial, constant, equation of a line

Key Formula

ax+b=0(one variable)ax+by=c(two variables)ax + b = 0 \quad \text{(one variable)} \qquad ax + by = c \quad \text{(two variables)}
Where:
  • x,yx, y = The unknown variables you solve for
  • a,ba, b = Coefficients — fixed numbers multiplying the variables
  • cc = A constant (a fixed number with no variable attached)

Worked Example

Problem: Solve the linear equation 3x + 7 = 22.
Step 1: Subtract 7 from both sides to begin isolating the variable term.
3x+7−7=22−7  ⟹  3x=153x + 7 - 7 = 22 - 7 \implies 3x = 15
Step 2: Divide both sides by 3 to solve for x.
3x3=153  ⟹  x=5\frac{3x}{3} = \frac{15}{3} \implies x = 5
Step 3: Check by substituting x = 5 back into the original equation.
3(5)+7=15+7=22  ✓3(5) + 7 = 15 + 7 = 22 \;\checkmark
Answer: x = 5

Another Example

This example has variables on both sides of the equation, requiring an extra step to collect like terms before solving.

Problem: Solve the linear equation 5a + 9 = 2a − 6.
Step 1: Subtract 2a from both sides so all variable terms are on the left.
5a−2a+9=−6  ⟹  3a+9=−65a - 2a + 9 = -6 \implies 3a + 9 = -6
Step 2: Subtract 9 from both sides to isolate the variable term.
3a=−6−9  ⟹  3a=−153a = -6 - 9 \implies 3a = -15
Step 3: Divide both sides by 3.
a=−153=−5a = \frac{-15}{3} = -5
Step 4: Check: substitute a = −5 into both sides of the original equation.
5(−5)+9=−25+9=−16and2(−5)−6=−10−6=−16  ✓5(-5) + 9 = -25 + 9 = -16 \quad\text{and}\quad 2(-5) - 6 = -10 - 6 = -16 \;\checkmark
Answer: a = −5

Frequently Asked Questions

What makes an equation linear?
An equation is linear if every variable in it has an exponent of exactly 1 and no two variables are multiplied together. Terms like x², xy, or √x would make the equation non-linear. Constants (plain numbers) are always allowed.
What is the difference between a linear equation and a linear function?
A linear equation is a statement that two expressions are equal, such as 2x + 3 = 11. A linear function is a rule that assigns each input exactly one output, often written as f(x) = mx + b. Every linear function can be described by a linear equation, but a linear equation like x = 4 does not define a function of x.
Can a linear equation have more than one variable?
Yes. A linear equation can have one variable (3x + 1 = 10), two variables (2x + 5y = 1), or even more. With two variables, the equation represents a line on the coordinate plane. With three variables, it represents a plane in three-dimensional space.

Linear Equation vs. Quadratic Equation

Linear EquationQuadratic Equation
Highest exponent on the variable1 (e.g., 3x + 2 = 8)2 (e.g., x² + 3x + 2 = 0)
General form (one variable)ax + b = 0ax² + bx + c = 0
Number of solutions (one variable)Exactly one (unless 0 = 0 or a contradiction)Up to two real solutions
Graph (two-variable form)Straight lineParabola
Solving methodIsolate the variable using inverse operationsFactoring, completing the square, or the quadratic formula

Why It Matters

Linear equations are one of the most frequently used tools in all of algebra. You will encounter them when modeling real-world situations — calculating costs, converting temperatures, predicting trends, and solving mixture or motion problems. Mastering them is also essential preparation for systems of equations, inequalities, and the study of linear functions and their graphs.

Common Mistakes

Mistake: Performing an operation on only one side of the equation.
Correction: Whatever you do to one side, you must do to the other. If you subtract 7 from the left side, subtract 7 from the right side as well. The equals sign means both sides must stay balanced.
Mistake: Confusing a linear equation with expressions that contain x², xy, or x inside a square root.
Correction: A linear equation requires every variable to appear to the first power only, with no variable-to-variable products. If you see x², 1/x, or √x, the equation is not linear.

Related Terms