Chapter focus
Algebra uses letters to represent unknown or changing numbers. Expressions describe calculations; equations state that two quantities are equal.
These original explanations and questions support revision. Follow the complete examples and exercises in your assigned Ganita Prakash Part II.
1. Expressions and like terms
In 5x + 3, x is a variable, 5 is its coefficient and 3 is a constant. The expression has no equality sign, so it is not an equation. If x = 4, its value is 5(4) + 3 = 23.
Like terms have the same variable part: 4a and 3a are like terms, so 4a + 3a = 7a. But 4a and 3b are unlike terms and cannot be combined. Constants can be combined with constants.
2. Simplify and substitute
Worked example: simplify 4a + 3a − 2 + 5
- Collect variable terms: 4a + 3a = 7a.
- Combine constants: −2 + 5 = 3.
- The simplified expression is 7a + 3.
- If a = 2, the value is 7(2) + 3 = 17.
Use brackets when substituting negative values: if x = −3, then x2 = (−3)2 = 9.
3. Equations represent balance
An equation says its left and right sides have equal values. To keep that equality true, perform the same operation on both sides. Undo addition or subtraction first, then multiplication or division as needed.
Worked example: solve 3x + 5 = 20
- Subtract 5 from both sides: 3x = 15.
- Divide both sides by 3: x = 5.
- Check in the original: 3(5) + 5 = 20, so the solution is correct.
Worked example: solve 5y − 7 = 18
Add 7 to both sides: 5y = 25. Divide by 5: y = 5. Check: 5(5) − 7 = 18.
4. Translate a word problem
Let an unknown number be x. “Three more than twice a number is 17” translates to 2x + 3 = 17. Subtract 3: 2x = 14. Divide by 2: x = 7. Check the statement: twice 7 is 14, and three more is 17.
Define the variable, form the equation from the words, solve one step at a time and check the value against the original situation.
Common errors to avoid
- 4x means 4 × x; it does not mean 4 + x.
- Only combine like terms: 2x + 3y cannot be simplified to 5xy.
- When solving an equation, preserve equality by doing the same operation on both sides.
- Check a solution in the original equation, not only in the last rearranged line.
Practice: write, simplify and solve
1. Simplify 4a + 3a − 2.
Answer: 7a − 2.
2. Solve 5y − 7 = 18 and check.
Answer: 5y = 25, so y = 5. Check: 25 − 7 = 18.
3. Evaluate 2m + 3 when m = 6.
Answer: 2(6) + 3 = 15.
4. A number increased by 9 is 31. Find the number.
Answer: Let it be n. n + 9 = 31, so n = 22.
5. Solve 4(x + 2) = 28.
Answer: Divide both sides by 4: x + 2 = 7; subtract 2: x = 5.
Chapter summary
Variables represent numbers, expressions describe operations and equations express equality. Simplify by collecting like terms, solve by preserving balance, and substitute the result into the original equation to check.