Expressions & Equations

Solving Linear Equations

Solving linear equations with variables on both sides.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Slope & Linear Relationships

Equations with the Variable on Both Sides

Before reading on, predict how many solutions each of these equations has: 2x + 3 = x + 7, and 2x + 3 = 2x + 7, and 2(x + 3) = 2x + 6. One has exactly one solution — but do you think the other two also have exactly one, or something different?

Solving 2x + 3 = x + 7 by subtracting x from both sides gives x + 3 = 7, so x = 4 — one solution, as expected. But subtracting 2x from both sides of 2x + 3 = 2x + 7 leaves 3 = 7 — a false statement no value of x can fix, meaning this equation has no solution at all. And simplifying 2(x + 3) = 2x + 6 by distributing gives 2x + 6 = 2x + 6 — a statement that's true no matter what x is, meaning every real number is a solution.

Definition — One Solution, No Solution, Infinitely Many Solutions

Solving a linear equation can lead to exactly one value of the variable (one solution), a false numerical statement with no variable left (no solution), or a true numerical statement with no variable left (infinitely many solutions) — depending on whether the variable terms fully cancel and, if so, what's left behind.

Worked Example — An Equation with No Solution

Solve 5x − 4 = 5x + 9. Subtract 5x from both sides: −4 = 9. This is false for any value of x, so the equation has no solution.

Worked Example — An Equation with Infinitely Many Solutions

Solve 3(x − 2) = 3x − 6. Distribute the left side: 3x − 6 = 3x − 6. Subtracting 3x from both sides gives −6 = −6, a true statement regardless of x — every real number satisfies this equation.

Equation Editor

Edit as raw LaTeX

sum(i, 1, n, i^2) = n(n+1)(2n+1)/6

Constants

Structures

Calculus & discrete math

Greek

\sum_{i=1}^{n} i^{2} = \frac{\operatorname{n}\left(n + 1\right) \cdot \left(2 \cdot n + 1\right)}{6}

Evaluate

Unknown function "n"

Tip

Whenever the variable completely cancels out while solving, look at what's left: a false statement means no solution, and a true statement means infinitely many solutions — the variable terms canceling is the signal to check which case applies.

Common Mistakes

Stopping partway through solving and reporting 'no solution' as soon as the variable cancels, without checking whether the remaining statement is true or false.
After the variable cancels, always evaluate the remaining numerical statement — a false one means no solution, but a true one means infinitely many solutions, a very different outcome.
Assuming every linear equation must have exactly one solution, the way most earlier equations did.
An equation with the variable on both sides can simplify to no solution or infinitely many solutions, not just one — check carefully rather than assuming.

Key Takeaways

A linear equation can have exactly one solution, no solution, or infinitely many solutions.
No solution occurs when the variable cancels, leaving a false numerical statement.
Infinitely many solutions occur when the variable cancels, leaving a true numerical statement.

Summary

Recognizing when an equation has no solution or infinitely many, not just one, completes single-equation solving. The next unit extends this to two equations at once, asking where their graphs meet.

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