Math · College algebra · Concept

Solving linear equations

To solve a linear equation, do the same operation to both sides until the variable stands alone. Clear any fractions, expand brackets, collect the variable terms on one side and divide by the coefficient.

What does it mean to solve an equation?

An equation claims that two expressions have the same value. A solution is a number that makes the claim true when it replaces the variable. In a linear equation the variable appears only to the first power: no x², no x under a root and no x in a denominator. Such an equation has exactly one solution, no solution, or every real number as a solution.

5 x - 7 = 2 x + 8

Why doing the same thing to both sides is safe

Adding or subtracting the same quantity on both sides, or multiplying or dividing both sides by the same nonzero number, gives an equivalent equation: one with exactly the same solutions. That is why every step can be trusted. Multiplying by zero is the exception, because 0 = 0 is true for every x and the information in the equation is lost.

a = b a + c = b + c, a = b k a = k b (k \neq 0)

\begin{array}{c} a = b a + c = b + c \\ a = b k a = k b (k \neq 0) \end{array}

Why must k be nonzero?

Multiplying x + 1 = 3 by 0 gives 0 = 0, which every number satisfies, so the new equation has more solutions than the old one. Dividing by zero is undefined. Either step breaks the chain of equivalent equations.

Collect the variable on one side

When x appears on both sides, subtract one of the x terms from both sides so the variable appears only once. Subtracting the smaller coefficient keeps the remaining coefficient positive, which avoids a sign slip. Then move the constant terms and divide by the coefficient.

\begin{aligned} 5 x - 7 & = 2 x + 8 \\ 3 x - 7 & = 8 \\ 3 x & = 15 \\ x & = 5 \end{aligned}

Clear fractions with the least common denominator

Multiply every term on both sides by the least common denominator (LCD) of the fractions. Each denominator then divides out, leaving whole-number coefficients. Put parentheses around a numerator with more than one term before you multiply, so a minus sign in front of the fraction reaches every term of it.

\frac{x}{3} + \frac{x - 1}{4} = 2 ⟹ 4 x + 3 (x - 1) = 24

\begin{array}{c} \frac{x}{3} + \frac{x - 1}{4} = 2 \\ ⟹ 4 x + 3 (x - 1) = 24 \end{array}

When the variable disappears

Sometimes the x terms cancel. If what remains is false, such as 6 = 5, no number makes the equation true: there is no solution. If what remains is always true, such as 6 = 6, every real number is a solution and the equation is an identity. Neither result means x = 0.

\begin{aligned} 2 (x + 3) & = 2 x + 5 \Rightarrow 6 = 5 \\ 2 (x + 3) & = 2 x + 6 \Rightarrow 6 = 6 \end{aligned}

What the last line tells you
Last line	Meaning
x = 10/3	One solution
6 = 5 (false)	No solution
6 = 6 (true)	Every real number is a solution

Check by substituting into the original equation

Substitute the answer into the original equation, not into a later line: a slip in an early step carries into every later line and still looks consistent there. Both sides must give the same number.

Common mistakes

Multiplying only the fractions by the LCD: every term on both sides, including whole numbers, must be multiplied.
Distributing a minus sign to only the first term: −3(x + 2) is −3x − 6, not −3x + 6.
Reading 0 = 0 as “x = 0”: it means every real number is a solution.
Dividing both sides by an expression containing x, which can divide by zero or lose a solution.

Key terms

Linear equation: An equation in which the variable appears only to the first power: not squared, not under a root and not in a denominator. In one variable it has one solution, no solution, or every real number as a solution.
Equivalent equations: Equations with exactly the same solution set. Adding the same quantity to both sides, or multiplying both sides by the same nonzero number, produces an equivalent equation.
Least common denominator: The smallest expression that every denominator in a problem divides into evenly. Multiplying an equation by it clears the fractions; for rational expressions it is built from the factored denominators.
Solution set: All the allowed values that make an equation or inequality true, such as {−2, 3}. Equivalent equations have the same solution set.
Mathematical identity: An equation that is true for every allowed value of the variable, such as (x + 1)² = x² + 2x + 1. Testing a few numbers can disprove an identity but can’t prove one.
Coefficient: The number or constant multiplying the variable part of a term, such as 5 in 5x². The leading coefficient multiplies a polynomial’s highest-degree term.

Work through an example

Solve (2x − 1)/3 − (x + 2)/4 = x/6, then check the answer.

Solve a linear equation with fractions →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

OpenStax College Algebra 2e — Linear equations in one variable

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Check each step in Math Graph both sides in Graph Open worked example on a board Isolating a variable in Math Reference

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