Introduction to Algebra (Copy)
Practice Questions — 2.1 Introduction to Algebra
Question 1
Simplify:
(a) 3x + 7x
(b) 5a − 2a + 9a
(c) 12y − 8y + 4y
Question 2
Substitute x = 4, y = −2 into each of the following:
(a) 3x + 2y
(b) 2x² − y
(c) xy − 5x
Question 3
If a = 5 and b = −3, evaluate:
(a) a² + 2ab + b²
(b) 3a − 4b
(c) ab − a²
Question 4
Simplify:
(a) 4(2x + 3)
(b) 5y − 2(3y − 4)
(c) 7a − (2a − 9)
Question 5
Expand and simplify:
(a) (x + 3)(x + 5)
(b) (2a − 4)(3a + 1)
(c) (y + 6)(y − 2)
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 6
Substitute p = 2, q = 3 into each of the following:
(a) 4p² − q²
(b) 2pq + q²
(c) p²q − 3p
Question 7
Simplify:
(a) 8m + 12n − 3m + 7n
(b) 10x − (4x + 7)
(c) 9y + 6 − (2y − 3)
Question 8
Expand and simplify:
(a) (x + 7)(x − 4)
(b) (3m + 5)(2m − 1)
(c) (2y − 3)(y − 5)
Question 9
If x = −1 and y = 5, evaluate:
(a) 2x² + 3y
(b) 4xy − y²
(c) (x − y)²
Question 10
Simplify fully:
(a) (2x + 3) + (5x − 7)
(b) (4a − 3) − (2a + 5)
(c) (6y − 4) + (y + 9)
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Answer key and explanations — 2.1 Introduction to Algebra
1.
(a) 3x + 7x = 10x
(b) 5a − 2a + 9a = 12a
(c) 12y − 8y + 4y = 8y
Explanation: Combine like terms (same variable, same power).
2. (x = 4, y = −2)
(a) 3(4) + 2(−2) = 12 − 4 = 8
(b) 2(4²) − (−2) = 2(16) + 2 = 32 + 2 = 34
(c) (4)(−2) − 5(4) = −8 − 20 = −28
Explanation: Substitute carefully; square applies before multiplication.
3. (a = 5, b = −3)
(a) a² + 2ab + b² = 25 + 2(5)(−3) + 9 = 25 − 30 + 9 = 4
(b) 3a − 4b = 15 − (−12) = 15 + 12 = 27
(c) ab − a² = (5)(−3) − 25 = −15 − 25 = −40
Explanation: Always apply correct order: squares, then multiplication, then addition/subtraction.
4.
(a) 4(2x + 3) = 8x + 12
(b) 5y − 2(3y − 4) = 5y − 6y + 8 = −y + 8
(c) 7a − (2a − 9) = 7a − 2a + 9 = 5a + 9
Explanation: Apply distributive property, watch for negative signs.
5.
(a) (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
(b) (2a − 4)(3a + 1) = 6a² + 2a − 12a − 4 = 6a² − 10a − 4
(c) (y + 6)(y − 2) = y² − 2y + 6y − 12 = y² + 4y − 12
Explanation: Expand double brackets using distributive/FOIL method.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
6. (p = 2, q = 3)
(a) 4p² − q² = 4(4) − 9 = 16 − 9 = 7
(b) 2pq + q² = 2(2)(3) + 9 = 12 + 9 = 21
(c) p²q − 3p = (4)(3) − 6 = 12 − 6 = 6
Explanation: Replace variables, then simplify.
7.
(a) 8m + 12n − 3m + 7n = (8m − 3m) + (12n + 7n) = 5m + 19n
(b) 10x − (4x + 7) = 10x − 4x − 7 = 6x − 7
(c) 9y + 6 − (2y − 3) = 9y + 6 − 2y + 3 = 7y + 9
Explanation: Watch signs when brackets are preceded by minus.
8.
(a) (x + 7)(x − 4) = x² − 4x + 7x − 28 = x² + 3x − 28
(b) (3m + 5)(2m − 1) = 6m² − 3m + 10m − 5 = 6m² + 7m − 5
(c) (2y − 3)(y − 5) = 2y² − 10y − 3y + 15 = 2y² − 13y + 15
Explanation: Expand carefully term by term.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
9. (x = −1, y = 5)
(a) 2x² + 3y = 2(1) + 15 = 17
(b) 4xy − y² = 4(−1)(5) − 25 = −20 − 25 = −45
(c) (x − y)² = (−1 − 5)² = (−6)² = 36
Explanation: Negative numbers must be squared with brackets.
10.
(a) (2x + 3) + (5x − 7) = 7x − 4
(b) (4a − 3) − (2a + 5) = 2a − 8
(c) (6y − 4) + (y + 9) = 7y + 5
Explanation: Remove brackets, combine like terms.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course