Solving Linear Equations (Basic)

A linear equation in one variable is an equation that can be written in the form ax + b = c, where a, b, and c are real numbers and a ≠ 0. The variable x appears to the first power only — no x², no x³, no square roots of x. The goal is always the same: find the value of x that makes the equation true. This value is called the solution or root of the equation. The fundamental principle behind solving any linear equation is the balance metaphor: an equation is like a perfectly balanced scale. Whatever operation you perform on one side, you must perform the exact same operation on the other side to keep the balance. Using the properties of equality — adding, subtracting, multiplying, or dividing both sides by the same nonzero quantity — you systematically isolate the variable until you have x = (some number). Linear equations in one variable appear everywhere in mathematics and science: calculating unknown distances, finding break-even points in economics, determining temperatures in physics, and solving mixture problems in chemistry. Mastering the technique of solving them is one of the most important foundational skills in all of algebra, and every more advanced topic — systems of equations, quadratics, calculus — builds directly on this skill.

Key takeaways

Simplify Each Side Separately
Move Variable Terms to One Side
Move Constant Terms to the Other Side
Isolate the Variable by Dividing (or Multiplying)
Check Your Solution

Simplify Each Side Separately

Before doing anything across the equals sign, simplify each side of the equation on its own. Distribute any parentheses using the distributive property: a(b + c) = ab + ac. Then combine like terms — terms that have the same variable factor. For example, 3x + 5x simplifies to 8x, and 7 − 3 simplifies to 4. Do not mix terms from opposite sides at this stage.

Move Variable Terms to One Side

Use the Addition or Subtraction Property of Equality to get all terms containing the variable onto one side of the equation. If the variable appears on both sides, add or subtract the smaller variable term from both sides. For instance, if you have 5x − 2 = 3x + 8, subtract 3x from both sides to get 2x − 2 = 8. It does not matter which side you choose, but moving the variable to the left is conventional.

Move Constant Terms to the Other Side

Use the Addition or Subtraction Property of Equality again to isolate the variable term. Any constant on the same side as the variable must be moved to the opposite side by adding or subtracting it from both sides. Continuing the example: 2x − 2 = 8 becomes 2x = 10 after adding 2 to both sides.

Isolate the Variable by Dividing (or Multiplying)

Once the equation has the form ax = c, divide both sides by the coefficient a (or multiply both sides by its reciprocal 1/a) to isolate x completely. In the example, divide both sides of 2x = 10 by 2 to get x = 5. If the coefficient is a fraction, multiplying both sides by its reciprocal is often cleaner.

Check Your Solution

Substitute the value you found back into the original equation. Evaluate both sides independently and confirm they are equal. For x = 5 in 5x − 2 = 3x + 8: left side = 5(5) − 2 = 23; right side = 3(5) + 8 = 23. Since 23 = 23 ✓, the solution is verified. This step catches arithmetic errors and is always worth doing.

The Key Equation

ax + b = c \quad \Longrightarrow \quad x = \frac{c - b}{a}, \quad a \neq 0

x — Variable (unknown) (dimensionless): The unknown quantity whose value we are solving for; it appears to the first power only.
a — Coefficient of x (dimensionless): The real number multiplying the variable x; must be nonzero for the equation to be linear and uniquely solvable.
b — Constant term on left (dimensionless): A real number added to or subtracted from the variable term on the left-hand side of the equation.
c — Constant term on right (dimensionless): A real number on the right-hand side of the equation; represents the value that ax + b equals.

Worked Example

Problem. Solve for x: 3(2x − 4) + 5 = 2x + 9

x = 4

Answer. x = 4

Figure

Diagram

Origins · 1 minAl-Khwarizmi and how this was discovered

Al-Khwarizmi

780–850 · Persian (Abbasid Caliphate)

Muhammad ibn Musa al-Khwarizmi wrote the treatise 'Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala' (The Compendious Book on Calculation by Completion and Balancing) around 820 CE. This work systematically described methods for solving linear and quadratic equations using the operations of 'al-jabr' (restoring, i.e., moving a subtracted term to the other side) and 'al-muqabala' (balancing, i.e., combining like terms). The word 'algebra' is derived directly from 'al-jabr.' Al-Khwarizmi presented solutions as step-by-step procedures — essentially the same balance-scale method taught in schools today. The word 'algorithm' comes from the Latinisation of al-Khwarizmi's own name — 'Algoritmi' — because his systematic procedures became so influential in medieval European mathematics that his name became synonymous with any step-by-step calculation procedure.

Linear Equations in Two Variables — The natural extension of one-variable equations; instead of a single unknown, you work with two, and solutions become lines on the coordinate plane.
Solving Linear Inequalities — Uses almost identical algebraic steps but produces a range of solutions rather than a single value; introduces the special rule about flipping the inequality symbol when multiplying or dividing by a negative.
Systems of Linear Equations — Combines multiple linear equations that must be satisfied simultaneously; substitution and elimination methods both reduce the system to a single linear equation in one variable.
Properties of Equality — The Addition, Subtraction, Multiplication, and Division Properties of Equality are the formal justification for every step taken when solving a linear equation.