In this chapter, you will learn
- —Understand equations and the concept of variables
- —Solve linear equations using transposition method
- —Solve equations using balance method
- —Translate word problems into equations
- —Verify solutions and solve real-life applications
What is an Equation?
An equation is a mathematical statement with an equal sign (=) showing two expressions are equal.
Definition: An equation has three parts: Left Side = Right Side. Variables (usually x, y, z) represent unknown values.
Example: 3x + 5 = 14 (x is the variable)
Types of Equations:
- Linear Equation: Highest power of variable is 1. Example: 2x + 3 = 7
- Quadratic Equation: Highest power of variable is 2. Example: x² + 2x = 0
Solution of an Equation: The value of the variable that makes the equation true is its solution.
- Example: In 2x + 3 = 7, the solution is x = 2 (because 2(2) + 3 = 7)
- Always verify: Substitute the solution back into the original equation
Exam Tip
Verify every solution by substituting back. This is essential in exams as it shows your understanding.
Common Mistake
Students confuse the equation with the solution. An equation is a statement; the solution is the value.
Solving by Transposition Method
Transposition means moving a term from one side to the other side of the equation.
Golden Rule: When we move a term across the equal sign, its sign changes.
+ changes to −, − changes to +, × changes to ÷, ÷ changes to ×
Steps to Solve:
- Write the equation clearly
- Move all terms with variables to left side
- Move all constant terms to right side
- Simplify both sides
- Solve for the variable
Example: Solve 2x + 5 = 13
2x + 5 = 13
2x = 13 − 5 (move 5 to right, sign changes)
2x = 8
x = 8 ÷ 2 (move 2 to right, ÷)
x = 4
Exam Tip
Master transposition—it's the fastest method for exams. Always check signs when moving terms.
Common Mistake
Forgetting to change signs during transposition. Remember: +5 becomes −5 when moved to the other side.
Solving by Balance Method
The balance method treats equations like a balance scale. Do the same operation on both sides to keep it balanced.
Key Principle: Whatever you do to one side, do to the other side to maintain equality.
Steps:
- Identify what needs to be removed from the variable side
- Apply the opposite operation to both sides
- Simplify
- Repeat until variable is alone
Example: Solve x ÷ 2 = 5
x ÷ 2 = 5
Multiply both sides by 2
(x ÷ 2) × 2 = 5 × 2
x = 10
Example with subtraction: Solve x − 3 = 7
x − 3 = 7
Add 3 to both sides
x − 3 + 3 = 7 + 3
x = 10
Exam Tip
Balance method is intuitive and easier to verify. Good for understanding; use transposition for speed.
Common Mistake
Forgetting to apply the operation to both sides. The equation must remain balanced.
Word Problems and Variables
Convert real-life situations into equations. This is the bridge between abstract math and practical problems.
Steps to Solve Word Problems:
- Read carefully and identify the unknown (choose a variable)
- Write the equation based on the given information
- Solve the equation
- Check if the answer makes sense in the context
Example: Ravi's age is 5 years more than Arjun's. If Ravi is 18, find Arjun's age.
Let Arjun's age = x
Ravi's age = x + 5 = 18
x = 18 − 5
x = 13 (Arjun is 13 years old)
Common Phrases in Word Problems:
- "more than" → add (+)
- "less than" → subtract (−)
- "times" → multiply (×)
- "divided by" → divide (÷)
Exam Tip
Always define the variable first. Write the equation before solving. Check answers in the original problem.
Common Mistake
Not reading the question carefully. Verify the answer makes sense (age can't be negative, etc.)
Verification and Algebraic Equations
Verification means checking if the solution is correct by substituting it back into the original equation.
How to Verify:
- Take the original equation
- Substitute the solution value for the variable
- Evaluate both sides
- Check if both sides are equal
Example: Verify x = 4 is solution of 2x + 5 = 13
Original: 2x + 5 = 13
Substitute x = 4: 2(4) + 5 = 13
Simplify: 8 + 5 = 13
Check: 13 = 13 ✓ (Verified!)
Why Verify?
- Ensures you didn't make calculation errors
- Confirms the solution is correct
- Shows complete understanding to examiners
- Catches mistakes before final answer
Exam Tip
Always write 'Verification:' and show substitution. It boosts confidence and ensures correctness.
Common Mistake
Skipping verification. Even if your method is correct, calculation errors can sneak in.
Chapter Summary
Simple Equations are linear expressions set equal to a value. Key learning points:
- Understanding Equations: Variables represent unknowns; solutions make equations true
- Transposition Method: Move terms across equals, changing signs (fastest for exams)
- Balance Method: Do same operation on both sides (best for understanding)
- Word Problems: Convert situations to equations using variables and operations
- Verification: Always substitute solution back into original equation
- Applications: Age, distance, cost problems solved using linear equations
Exam Focus: Solving equations (transposition), word problems, verification, algebraic manipulations.