Chapter Overview
An equation states that two expressions have the same value. Solving an equation means finding the value that keeps this balance true.
This chapter belongs to Ganita Prakash, Grade 7. Focus on explaining each step, checking whether an answer is reasonable, and comparing more than one solution method.
Learning Objectives
After studying this chapter, you should be able to:
- explain and apply expressions and equations;
- explain and apply balance principle;
- explain and apply inverse operations;
- explain and apply verification;
- communicate the reasoning behind a solution clearly;
- check answers using estimation, substitution, or a second method.
Key Concepts
| Concept | Meaning |
|---|---|
| Expressions and equations | 3x + 2 is an expression; 3x + 2 = 17 is an equation because it contains an equality. |
| Balance principle | Doing the same valid operation to both sides preserves equality. |
| Inverse operations | Addition reverses subtraction and multiplication reverses division, allowing the variable to be isolated systematically. |
| Verification | Substitute the proposed solution into the original equation and check that both sides match. |
Detailed Explanation
Expressions and equations
3x + 2 is an expression; 3x + 2 = 17 is an equation because it contains an equality.
Balance principle
Doing the same valid operation to both sides preserves equality.
Inverse operations
Addition reverses subtraction and multiplication reverses division, allowing the variable to be isolated systematically.
Verification
Substitute the proposed solution into the original equation and check that both sides match.
Do not memorise a rule without testing it on examples. Ask what each number, operation, line, or symbol represents and whether the result fits the original situation.
Important Rules and Formulae
- Maintain equality by changing both sides, not only one.
- Simplify each side before isolating the variable.
- State the solution and verify it in the original equation.
Worked Examples
Example 1
Problem: Solve 4x + 7 = 31.
Solution: Subtract 7: 4x = 24. Divide by 4: x = 6. Check: 4×6+7=31.
Example 2
Problem: A number divided by 5 and then increased by 3 gives 11. Find it.
Solution: x/5 + 3 = 11; subtract 3 to get x/5=8; multiply by 5, so x=40.
Show the mathematical statement, substitution or construction step before writing the final answer. Include units wherever the quantity is measured.
Common Mistakes
- Moving a term across the equals sign without understanding the inverse operation.
- Dividing only one term on a side containing a sum.
- Failing to check a negative or fractional answer.
Quick Revision
- Expressions and equations:
3x + 2is an expression;3x + 2 = 17is an equation because it contains an equality. - Balance principle: Doing the same valid operation to both sides preserves equality.
- Inverse operations: Addition reverses subtraction and multiplication reverses division, allowing the variable to be isolated systematically.
- Verification: Substitute the proposed solution into the original equation and check that both sides match.
Practice Questions
- Solve
x + 18 = 43. - Solve
7x = 63. - Solve
3x − 5 = 16. - Is
x = 4a solution of2x + 1 = 9?
Answers and Explanations
x = 25.x = 9.x = 7.- Yes.
Self-Check
Explain one rule from this chapter in your own words, create a fresh example, solve it, and verify the answer. If the explanation and verification agree, the concept is understood rather than merely memorised.