Expressions & Equations

Solving One-Variable Equations

Solving equations of the form x + p = q and px = q.

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

Prerequisites

Simplifying Expressions

Finding the Value That Makes It True

'A number plus 3 is 10.' Before reading on, find that number in your head — then think about what operation you actually did to find it, without writing anything down.

Most people mentally subtract 3 from 10 to undo the addition. That instinct is exactly the strategy behind solving equations formally: whatever operation connects the variable to a number, apply its inverse operation to both sides to isolate the variable — keeping the equation balanced the entire time.

Worked Example — Solving an Equation of the Form x + p = q

Solve n + 8 = 15. Undo the addition by subtracting 8 from both sides: n + 8 − 8 = 15 − 8, giving n = 7. Check: 7 + 8 = 15. True.

Worked Example — Solving an Equation of the Form px = q

Solve 9n = 63. Undo the multiplication by dividing both sides by 9: 9n ÷ 9 = 63 ÷ 9, giving n = 7. Check: 9 × 7 = 63. True.

Worked Example — Solving a Real-World Equation

A parking garage charges a flat $5 entry fee plus $2 per hour, and a customer's total bill was $17. How many hours did they park? Let h be the number of hours: 2h + 5 = 17. Subtract 5 from both sides: 2h = 12. Divide both sides by 2: h = 6. The customer parked for 6 hours.

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

Write out the check step every time, even for an equation solved mentally — substituting the answer back into the original equation catches mistakes before they carry into a larger problem.

Common Mistakes

Applying the inverse operation to only one side of the equation, breaking the balance between both sides.
Whatever operation undoes the variable's attachment must be applied to both sides of the equation — this is what keeps the equation true at every step.
Setting up a real-world equation with the constant and variable terms reversed, such as writing 5h + 2 = 17 for the parking example instead of 2h + 5 = 17.
Match each piece of the situation to its correct role: the amount that changes with hours (the rate) multiplies the variable, and the flat fee stands alone as a constant.

Key Takeaways

Solving an equation means finding the value of the variable that makes both sides equal.
Applying an inverse operation to both sides isolates the variable while keeping the equation balanced.
Writing a real-world situation as an equation means matching each described quantity to its correct algebraic role.

Summary

Solving equations formalizes the same undo-the-operation instinct used to solve simple problems mentally. The final lesson in this unit represents situations where many values work, not just one — using inequalities.

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