Solving simple equations comes down to one idea: undo whatever has been done to the variable, and do the same thing to both sides. Once a student sees that, x + 3 = 5 stops being a puzzle and becomes a two-second job.
This guide shows how to solve simple equations step by step, from one-step equations to brackets and fractions. Every example comes with its solution and a check, so your child can follow along and test themselves.
What Is a Simple Equation? The Variable, the Equals Sign and Both Sides
A simple equation is an equation with one variable, usually x, raised to the first power. The equals sign splits it into a left side and a right side, also called the left-hand side (LHS) and right-hand side (RHS), and both sides have the same value.
In x + 3 = 5, the unknown is x, 3 is a constant, and the equation is true only when x = 2. To solve simple equations means finding that one value.
Keep It Balanced: The Golden Rule of Algebra
Think of an equation as a balance scale. The golden rule of algebra says that whatever you do to one side, you must do to the other, so the scale stays balanced.
You may add, subtract, multiply, or divide both sides by the same number (except dividing by zero), as long as you apply the same operation to each side. Each move produces equivalent equations: new equations with the same solution. That rule is the core of the solving process in algebra, from Secondary 1 to university.
Some textbooks call the shortcut “transposition”: moving a term to the other side and flipping its sign. Transposition is the golden rule written faster; −10 on the left becomes +10 on the right because you added 10 to both sides.
Where Simple Equations Fit: Linear Equations vs Quadratic Equations
Simple equations are the easiest linear equations: the variable appears only to the first power, so there is exactly one solution. A simple linear equation such as 2x + 4 = 10 graphs as a straight line, which is where the name linear equation comes from. Each step on this page works for any first-degree equation.
Equations where the variable is squared, like x² − 5x + 6 = 0, are quadratic equations. They can have two solutions and need different methods, covered in our guide to the quadratic equation. Systems with two variables, such as x + y = 10 and x − y = 2, are also a separate topic.
Solving Simple Equations with Inverse Operations
Solving simple equations relies on inverse operations, pairs of operations that cancel each other. Addition and subtraction undo each other; multiplication and division undo each other.
Mathematicians name the tools: subtracting 3 adds the additive inverse of 3, and dividing by 3 multiplies by the multiplicative inverse, 1/3. Students only need the plain version: find what was done to x and do the opposite operation. The goal of every step is to isolate the variable, leaving x alone on one side.
One-Step Equations: Add, Subtract, Multiply, or Divide
One-step equations need a single inverse operation to get the variable by itself:
| Equation | Inverse operation | Solution |
|---|---|---|
| x + 3 = 5 | Subtract 3 from both sides | x = 2 |
| x − 4 = 9 | Add 4 to both sides | x = 13 |
| 3x = 12 | Divide both sides by 3 | x = 4 |
| x/5 = 7 | Multiply both sides by 5 | x = 35 |
Each answer can be checked in a second. Put x = 4 into 3x = 12: 3 × 4 = 12, which is true.
In 3x, the 3 is a coefficient, a number multiplying the variable. Dividing by the coefficient is the most common final step in any equation.
Two-Step Equations: Undo the Addition, Then the Multiplication
Two-step equations combine two operations, so you undo them in reverse. Working backwards through the order of operations (BODMAS), you undo addition and subtraction first, then multiplication and division.
Solve 2x + 4 = 10:
- Subtract 4 from both sides: 2x = 6
- Divide both sides by 2: x = 3
- Check: 2(3) + 4 = 10, so the solution is correct
The same steps solve 3x − 5 = 10. Add 5 to get 3x = 15, then divide by 3: x = 5.
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How to Solve an Equation with Brackets, Fractions or Variables on Both Sides
How to solve an equation that looks harder? Simplify each side first, then use the same inverse operations. Brackets, fractions, and an x on both sides only mean one extra tidy-up step before the usual two.
The Distributive Property and Like Terms
The distributive property removes brackets (also called parentheses): a(b + c) = ab + ac. Expand first, then solve.
Solve 4(x + 2) = 20. Expand to 4x + 8 = 20, subtract 8 to get 4x = 12, and divide by 4: x = 3.
Like terms are terms with the same variable, such as 2x and 3x. Combine like terms before anything else: 2x + 3x − 4 = 11 simplifies to 5x − 4 = 11, so 5x = 15 and x = 3.
Equations with Variables on Both Sides
When x appears on both sides, collect the variable terms on one side first. Subtract the smaller x term to avoid negative coefficients.
Solve 4x + 2 = 2x + 10. Subtract 2x from both sides: 2x + 2 = 10. Subtract 2: 2x = 8. Divide by 2: x = 4. These multi-step problems are where simple equations meet full first-degree equations.
Fractions, Reciprocals and Negative Numbers
Fractions disappear when you multiply. For x/3 + 2 = 6, subtract 2 to get x/3 = 4, then multiply by 3: x = 12.
A fraction coefficient needs its reciprocal. In (2/3)x = 8, multiply both sides by 3/2, so x = 12.
Negative numbers trip students up more than any other detail. In −2x = 10, divide both sides by −2, not 2, so x = −5. The check confirms it: −2 × −5 = 10.
Solve for x, Then Check the Value in the Original Equation
Every time you solve for x, substitute the value back into the original equation. If both sides come out equal, the answer is right; if not, one step went wrong.
Checking takes ten seconds and catches most errors before a teacher does. Tutors make it a habit from the very first lesson.
Younger students often use trial and error: guess a value, test it, adjust. That works for x + 3 = 5 but slows down fast on 7x − 13 = 50. The systematic method on this page, solving an equation with inverse operations, reaches x = 9 in two steps: add 13 to get 7x = 63, then divide by 7. Guessing is a fine way to check intuition; it is a poor way to solve.
Common Mistakes When You Solve Equations
The same errors show up in almost every notebook. These are the common mistakes to watch for:
- Doing an operation to one side only
- Dropping a negative sign when moving a term
- Dividing only part of an expression: (2x + 4) ÷ 2 is x + 2, not x + 4
- Forgetting to check the answer in the original equation
Most of these surface during homework help, when a student works alone and can’t tell where the answer went wrong.
Word Problems from Quebec Secondary 1 Math
In Quebec, Secondary 1 and Secondary 2 math turn sentences into equations. Here is a typical word problem: Maya buys 3 notebooks and a $4 pen and pays $19. How much is one notebook?
Let n be the price of a notebook. The equation is 3n + 4 = 19. Subtract 4: 3n = 15. Divide by 3: n = 5, so each notebook costs $5. The skill of solving math problems is mostly this translation step.
Students who fell behind on it can catch up in math before the gap grows.
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