How to Solve Linear Equations Using Addition and Subtraction Step by Step
Solving equations by using different operations such as addition and subtraction is a foundational skill in algebra that every student must master. This topic is essential not only for scoring well in school and competitive exams (like JEE or Olympiads) but also for tackling everyday problems involving unknowns—such as budgeting or splitting costs. Knowing how to isolate variables using addition or subtraction helps you solve a wide range of linear equations and builds confidence for more advanced topics.
What Does It Mean to Solve Equations by Addition and Subtraction?
An equation is a mathematical statement that shows two expressions are equal, usually separated by an equals sign ( = ). The aim of solving an equation is to find the value of the variable (like x or y) that makes the equation true. In many basic equations, the variable is combined with a number through addition or subtraction. By performing the opposite operation—subtracting if it’s addition, or adding if it’s subtraction—you can isolate the variable on one side and find its value.
For example, in the equation x + 5 = 9, you would subtract 5 from both sides to find the value of x. This technique also works in reverse for subtraction: if you have x − 4 = 10, you would add 4 to both sides.
Key Principles and Steps
- Variable: A symbol that represents an unknown value (e.g., x, y).
- Equation: A statement showing two expressions are equal (e.g., x + 3 = 7).
- Balancing Method: Any operation you do to one side of the equation, you must also do to the other side to keep it balanced.
- Inverse Operation: The action that “undoes” the effect of another operation (addition ↔ subtraction).
The core idea is: "Add or subtract the same value from both sides to keep the equation balanced."
How to Solve Linear Equations Using Addition and Subtraction
- Identify the operation attached to the variable (addition or subtraction).
- Perform the inverse operation on both sides of the equation.
- Simplify both sides to solve for the variable.
- Substitute the answer back into the original equation to check your solution.
Examples of Solving Equations by Addition and Subtraction
Example 1: Solve for x in the equation x + 7 = 15
- Identify the operation: Addition (+7).
- Subtract 7 from both sides: x + 7 − 7 = 15 − 7
- Simplify: x = 8
- Check: 8 + 7 = 15 (Correct!)
Example 2: Solve for x in the equation x − 5 = 12
- Identify the operation: Subtraction (–5).
- Add 5 to both sides: x − 5 + 5 = 12 + 5
- Simplify: x = 17
- Check: 17 − 5 = 12 (Correct!)
Example 3: Word Problem
"Four more than a certain number is 19. Find the number."
- Let the number be x.
- Equation: x + 4 = 19
- Subtract 4 from both sides: x = 15
Practice Problems
- Solve for x: x + 6 = 20
- Solve for x: x − 8 = 13
- Solve for y: y + 11 = 25
- Solve for n: n − 17 = 22
- Tom withdraws ₹120 from his bank account. His remaining balance is ₹320. What was his original balance?
- Three less than a number is 11. What is the number?
Common Mistakes to Avoid
- Not performing the operation on both sides of the equation.
- Reversing the operation (adding when you should subtract, or vice versa).
- Dropping negative signs or copying numbers incorrectly.
- Forgetting to check your answer by substituting it back into the original equation.
Real-World Applications
- Splitting bills at restaurants or among friends.
- Finding out original amounts after withdrawals/deposits from bank accounts.
- Shopping discounts or budgeting (e.g., "I had x rupees, spent 120, now I have 700. How much did I start with?")
- Simple physical problems like measuring remaining distance or time left.
These scenarios show why mastering basic equation-solving is a life skill, not merely a math classroom requirement.
Related Topics to Explore
- Solving Equations With Variables on Both Sides
- Addition and Subtraction of Algebraic Expressions
- Algebraic Expression
- Simple Equations
In this topic, you learned how to solve equations by using addition and subtraction—key operations in algebra. Remember, always perform the inverse operation and keep both sides of the equation balanced. These techniques, taught simply at Vedantu, will strengthen your problem-solving skills for school, competitive exams, and daily life situations.
FAQs on Solving Equations by Using Addition and Subtraction Methods
1. What does it mean to solve an equation using addition and subtraction?
To solve an equation using addition and subtraction means to isolate the variable by adding or subtracting the same number on both sides of the equation. This keeps the equation balanced and helps find the value of the unknown.
- Use addition to cancel subtraction.
- Use subtraction to cancel addition.
- Always perform the same operation on both sides.
2. How do you solve a simple equation using subtraction?
You solve a simple equation using subtraction by subtracting the same number from both sides to isolate the variable. This maintains equality and removes the added number.
- Example: x + 9 = 15
- Subtract 9 from both sides.
- x + 9 − 9 = 15 − 9
- Simplify to get x = 6.
3. How do you solve an equation using addition?
You solve an equation using addition by adding the same number to both sides to cancel out subtraction. This helps isolate the unknown variable.
- Example: x − 4 = 10
- Add 4 to both sides.
- x − 4 + 4 = 10 + 4
- Simplify to get x = 14.
4. What is the addition and subtraction property of equality?
The addition and subtraction property of equality states that adding or subtracting the same number from both sides of an equation keeps the equation balanced. In symbols:
- If a = b, then a + c = b + c
- If a = b, then a − c = b − c
5. Can you give an example of solving a one-step equation?
A one-step equation can be solved by performing a single addition or subtraction to isolate the variable. For example:
- Equation: x − 7 = 3
- Add 7 to both sides.
- x − 7 + 7 = 3 + 7
- Simplify to get x = 10.
6. How do you solve equations with variables on both sides using addition and subtraction?
To solve equations with variables on both sides, use addition or subtraction to move all variable terms to one side and constants to the other. This simplifies the equation into a one-step form.
- Example: 3x + 5 = x + 13
- Subtract x from both sides: 2x + 5 = 13
- Subtract 5 from both sides: 2x = 8
- Divide by 2: x = 4.
7. Why must you perform the same operation on both sides of an equation?
You must perform the same operation on both sides of an equation to keep it balanced and maintain equality. An equation represents two equal expressions, so changing only one side breaks the equality.
- Example: If x + 3 = 8
- Subtracting 3 from both sides keeps it equal.
- If you subtract from one side only, the equation becomes incorrect.
8. What are common mistakes when solving equations using addition and subtraction?
Common mistakes when solving equations include not applying the operation to both sides and making sign errors. These mistakes can lead to incorrect solutions.
- Forgetting to change the sign when adding or subtracting.
- Applying the operation to only one side.
- Combining unlike terms incorrectly.
9. How do you check your solution to an equation?
You check your solution by substituting the found value back into the original equation to see if both sides are equal. If both sides match, the solution is correct.
- Example: If x = 6 for x + 4 = 10
- Substitute: 6 + 4 = 10
- 10 = 10 ✔
10. What is the difference between one-step and two-step equations?
The difference between one-step and two-step equations is that one-step equations require only one operation, while two-step equations require two operations to isolate the variable.
- One-step example: x + 5 = 9 → x = 4
- Two-step example: 2x + 3 = 11
- Subtract 3: 2x = 8
- Divide by 2: x = 4
