What is an equation?
An equation is a mathematical sentence that shows two expressions are equal. An equation always includes an equal sign (=). Think of it as a balanced scale: whatever is on the left side must have the same value as whatever is on the right side.
- 5 + 2 = 7
- 10 = 6 + 4
- 3 × 4 = 12
The equal sign does not mean “the answer is coming.” It means “the same as.” Both sides of an equation must always be equal in value.
What is a variable?
A variable is a symbol, usually a letter, that stands for an unknown number. In fourth grade, we often use letters like x, y, or n to represent the value we need to find. When you solve an equation, your goal is to figure out what number the variable represents.
- In the equation x + 4 = 9, the variable x is the unknown.
- In the equation 15 - y = 8, the variable y is the unknown.
- In the equation 6 × n = 24, the variable n is the unknown.
A variable can be any letter. It does not change the math. You can think of the variable as a mystery number waiting to be discovered.
Solving equations: the balance method
To solve an equation means to find the value of the variable that makes the equation true. The balance method is a powerful strategy: whatever you do to one side of the equation, you must do to the other side to keep it balanced. Your goal is to get the variable alone on one side of the equal sign.
- Equation: x + 3 = 10
To isolate x, subtract 3 from both sides: x + 3 - 3 = 10 - 3 → x = 7 - Equation: y - 5 = 12
To isolate y, add 5 to both sides: y - 5 + 5 = 12 + 5 → y = 17 - Equation: 4 × a = 20
To isolate a, divide both sides by 4: (4 × a) ÷ 4 = 20 ÷ 4 → a = 5
Use inverse operations—the opposite operation—to undo what is being done to the variable. Addition undoes subtraction, and multiplication undoes division.
Solving addition and subtraction equations
When a variable is in an addition or subtraction equation, you use the inverse operation to find its value. If the variable is added to a number, subtract that number from both sides. If the variable has a number subtracted from it, add that number to both sides. Always check your answer by substituting it back into the original equation.
- Solve: n + 12 = 25
Subtract 12 from both sides: n + 12 - 12 = 25 - 12 → n = 13
Check: 13 + 12 = 25 ✓ - Solve: 34 = m + 19
Subtract 19 from both sides: 34 - 19 = m + 19 - 19 → 15 = m
Check: 34 = 15 + 19 ✓ - Solve: k - 8 = 22
Add 8 to both sides: k - 8 + 8 = 22 + 8 → k = 30
Check: 30 - 8 = 22 ✓ - Solve: 45 = p - 16
Add 16 to both sides: 45 + 16 = p - 16 + 16 → 61 = p
Check: 45 = 61 - 16 ✓
Always perform the same operation on both sides. This keeps the equation balanced, just like a fair scale. Checking your work helps catch simple mistakes.
Solving multiplication and division equations
Multiplication and division equations are solved using inverse operations as well. If a variable is multiplied by a number, divide both sides by that number. If a variable is divided by a number, multiply both sides by that number. In fourth grade, you will work with whole numbers and basic facts.
- Solve: 7 × w = 56
Divide both sides by 7: (7 × w) ÷ 7 = 56 ÷ 7 → w = 8
Check: 7 × 8 = 56 ✓ - Solve: 48 = 6 × t
Divide both sides by 6: 48 ÷ 6 = (6 × t) ÷ 6 → 8 = t
Check: 48 = 6 × 8 ✓ - Solve: r ÷ 5 = 9
Multiply both sides by 5: (r ÷ 5) × 5 = 9 × 5 → r = 45
Check: 45 ÷ 5 = 9 ✓ - Solve: 11 = s ÷ 3
Multiply both sides by 3: 11 × 3 = (s ÷ 3) × 3 → 33 = s
Check: 11 = 33 ÷ 3 ✓
Remember that division by zero is not possible. In all our equations, the divisor will never be zero. Think of multiplication and division as a team—one undoes the other.
Equations with two operations (two-step equations)
Some equations require more than one step to solve. These are called two-step equations. You will first undo addition or subtraction, and then undo multiplication or division. Always follow the order of operations in reverse: undo addition or subtraction before undoing multiplication or division.
- Solve: 2x + 3 = 11
Step 1: Subtract 3 from both sides: 2x + 3 - 3 = 11 - 3 → 2x = 8
Step 2: Divide both sides by 2: 2x ÷ 2 = 8 ÷ 2 → x = 4
Check: 2 × 4 + 3 = 8 + 3 = 11 ✓ - Solve: 5y - 4 = 16
Step 1: Add 4 to both sides: 5y - 4 + 4 = 16 + 4 → 5y = 20
Step 2: Divide both sides by 5: 5y ÷ 5 = 20 ÷ 5 → y = 4
Check: 5 × 4 - 4 = 20 - 4 = 16 ✓ - Solve: (a ÷ 3) + 2 = 7
Step 1: Subtract 2 from both sides: (a ÷ 3) + 2 - 2 = 7 - 2 → a ÷ 3 = 5
Step 2: Multiply both sides by 3: (a ÷ 3) × 3 = 5 × 3 → a = 15
Check: (15 ÷ 3) + 2 = 5 + 2 = 7 ✓
When solving two-step equations, look for the number that is farthest from the variable and undo that operation first. This strategy simplifies the equation step by step.
Writing equations from word problems
Real-world problems often describe a situation that can be written as an equation. To write an equation, identify the unknown quantity and represent it with a variable. Then translate the words into numbers and operations. Finally, solve the equation to answer the question.
- Problem: Mia had some stickers. Her friend gave her 12 more stickers. Now Mia has 35 stickers. How many stickers did Mia start with?
Write: Let s = stickers Mia started with. The equation is s + 12 = 35.
Solve: s = 35 - 12 → s = 23. Mia started with 23 stickers. - Problem: A pizza was cut into 8 equal slices. After the party, 3 slices were left. The number of slices eaten was 5. Write an equation to show this.
Write: Let e = slices eaten. The equation is 8 - e = 3, or e + 3 = 8.
Solve: e = 5. Five slices were eaten. - Problem: Liam bought 4 identical notebooks for a total of $28. What is the cost of one notebook?
Write: Let c = cost of one notebook. The equation is 4 × c = 28.
Solve: c = 28 ÷ 4 → c = 7. Each notebook costs $7.
Key words like “total,” “more,” “less,” “times,” and “each” give clues about which operation to use. Reading carefully is the first step to writing the correct equation.
Checking solutions and using estimation
After you find a value for the variable, it is important to check that your answer makes sense. Substitute the value back into the original equation and verify that both sides are equal. You can also use estimation to see if your answer is reasonable before solving.
- Check: Solve x - 18 = 42 → x = 60.
Substitute: 60 - 18 = 42 → 42 = 42 ✓ Correct. - Estimation: Before solving 39 + y = 112, estimate: y is about 112 - 40 = 72. The actual solution is 73, which is close to the estimate.
- Check a word problem: If the answer to “How many apples?” is 12, but the problem said there were 25 apples total and 13 were given away, 25 - 13 = 12 makes sense.
Checking your solution is a habit of strong mathematicians. It takes only a few seconds and helps you catch errors before you consider your work complete.
Common mistakes and how to avoid them
When solving equations, there are a few common errors. Recognizing these mistakes can help you avoid them and solve with confidence. The most frequent issues include performing the wrong inverse operation, forgetting to perform the operation on both sides, and misreading the variable’s position.
- Mistake: x + 5 = 12 → x = 12 + 5 → x = 17 (Incorrect)
Correct: Subtract 5 from both sides: x + 5 - 5 = 12 - 5 → x = 7 - Mistake: 4y = 20 → y = 20 - 4 → y = 16 (Incorrect)
Correct: Divide both sides by 4: 4y ÷ 4 = 20 ÷ 4 → y = 5 - Mistake: In the equation 24 = m ÷ 3, solving as m = 24 ÷ 3 = 8 (Incorrect)
Correct: Multiply both sides by 3: 24 × 3 = (m ÷ 3) × 3 → m = 72
Always ask: “What operation is being done to the variable?” Then do the opposite on both sides. Drawing a vertical line through the equal sign can help you remember to keep both sides balanced.
Review of key vocabulary
Understanding the language of equations is essential for success. Here are the key terms you will use when solving equations with one variable. Review these words and their meanings to strengthen your math communication skills.
- Equation: A mathematical sentence with an equal sign.
- Variable: A symbol that represents an unknown number.
- Inverse Operation: Operations that undo each other (addition and subtraction; multiplication and division).
- Solve: To find the value of the variable that makes the equation true.
- Balance Method: The strategy of performing the same operation on both sides of an equation.
- Two-Step Equation: An equation that requires two inverse operations to solve.
- Check: The process of substituting the solution back into the original equation to verify it is correct.
Using correct vocabulary helps you explain your reasoning clearly. When you solve equations, try to use these words to describe what you are doing step by step.
Common Core alignment: CCSS.MATH.CONTENT.4.OA.A.3 – Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.OA.A.3. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework. Encourage students to verbalize the inverse operations and check their solutions. The balance method builds a strong foundation for algebraic thinking in later grades.