Mathematics for SGT and TET Paper 1A (Classes III-VIII) · Chapter 10

Algebra: Expressions, Identities, Linear Equations and Exponents

What to remember

An algebraic expression is made of terms joined by + or −. Only like terms can be added or subtracted. The degree of a polynomial is the highest power of the variable.
Key identities: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², a² − b² = (a + b)(a − b), and (x + a)(x + b) = x² + (a + b)x + ab.
A linear equation in one variable has the form ax + b = 0 (a ≠ 0) and has exactly one solution. Whatever is done to one side must be done to the other. Laws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1.

Variables, constants and expressions

A variable is a letter (x, y, n) that can take different values. A constant has a fixed value. A term is a number, a variable or a product of them, such as 5x, −3y², 7. The terms of an expression are separated by + or − signs. The number multiplying the variable is the coefficient: in 5x the coefficient is 5, and in −3y² it is −3.

Like terms have the same variables with the same powers: 3x² and −7x² are like; 3x² and 3x are unlike. Only like terms can be added or subtracted: 5x + 3x = 8x; 7a² − 2a² = 5a².

Type	Terms	Example
Monomial	1	7x, −5y², 9
Binomial	2	x + 4, 3a − 2b
Trinomial	3	x² + 2x + 1
Polynomial	One or more	x³ − 2x + 5

Degree: The degree of a term in one variable is its power. The degree of a polynomial is the highest power. x³ − 2x + 5 has degree 3. A constant (non-zero) has degree 0. In x²y³ the degree is 2 + 3 = 5. Degree 1 is linear, 2 is quadratic, 3 is cubic.

Operations on expressions:

Addition: (3x + 5) + (2x − 7) = 5x − 2.
Subtraction: change all signs of the second expression. (5a − 3) − (2a − 4) = 3a + 1.
Multiplication: multiply each term by each term. (x + 2)(x + 3) = x² + 5x + 6. A monomial times a monomial: (3x)(−4x²) = −12x³.
Order of signs: (+)(+) = +, (−)(−) = +, (+)(−) = −.

Value of an expression: Substitute the value. If x = 2, then 3x² − 5x + 1 = 12 − 10 + 1 = 3.

Algebraic identities

An identity is true for every value of the variable. An equation is true only for certain values.

Identity	Expanded form
(a + b)²	a² + 2ab + b²
(a − b)²	a² − 2ab + b²
a² − b²	(a + b)(a − b)
(x + a)(x + b)	x² + (a + b)x + ab
(a + b)³	a³ + 3a²b + 3ab² + b³

Using the identities:

102² = (100 + 2)² = 10000 + 400 + 4 = 10404.
98² = (100 − 2)² = 10000 − 400 + 4 = 9604.
103 × 97 = (100 + 3)(100 − 3) = 10000 − 9 = 9991.
52 × 48 = 50² − 2² = 2500 − 4 = 2496.
If a + b = 7 and ab = 10, then a² + b² = (a + b)² − 2ab = 49 − 20 = 29.
If x + 1/x = 5, then x² + 1/x² = 25 − 2 = 23.
(a + b)² − (a − b)² = 4ab and (a + b)² + (a − b)² = 2(a² + b²).

Linear equations in one variable

An equation says that two expressions are equal. A linear equation in one variable has only one variable, with power 1. Its solution (root) is the value that makes both sides equal.

Rules:

Add or subtract the same number on both sides.
Multiply or divide both sides by the same non-zero number.
Transposition: a term moves to the other side with its sign changed (+ becomes −, × becomes ÷).

Examples:

3x + 5 = 20 → 3x = 15 → x = 5.
2(x − 3) = 10 → x − 3 = 5 → x = 8.
x/3 + 2 = 6 → x/3 = 4 → x = 12.
5x − 7 = 3x + 9 → 2x = 16 → x = 8.
(3x + 1)/2 = 5 → 3x + 1 = 10 → x = 3.
Equations like (x + 1)/(x + 2) = 3/4 are cross-multiplied: 4x + 4 = 3x + 6 → x = 2.

Checking: Put the answer back. For x = 8 in 5x − 7 = 3x + 9: left = 33, right = 33.

Word problems — steps: (1) Read and choose a variable. (2) Write the equation. (3) Solve. (4) Check with the words of the problem.

Statement	Equation
A number increased by 7 is 20	x + 7 = 20
Twice a number minus 3 is 11	2x − 3 = 11
Sum of two consecutive numbers is 41	x + (x + 1) = 41
A number is 4 more than 3 times another	y = 3x + 4

Sum of three consecutive numbers is 72: x + (x + 1) + (x + 2) = 72 → 3x = 69 → x = 23. The numbers are 23, 24, 25.
A father is 3 times as old as his son, and the sum of their ages is 60. Then x + 3x = 60, so the son is 15 and the father is 45.
The perimeter of a rectangle is 60 cm and its length is twice the breadth: 2(2b + b) = 60 → b = 10, length 20 cm.

Exponents (powers)

aⁿ means a multiplied by itself n times. a is the base and n is the exponent (index).

Law	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	2³ × 2⁴ = 2⁷ = 128
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	5⁶ ÷ 5⁴ = 5² = 25
(aᵐ)ⁿ = aᵐⁿ	(3²)³ = 3⁶ = 729
aᵐ × bᵐ = (ab)ᵐ	2³ × 5³ = 10³ = 1000
aᵐ ÷ bᵐ = (a/b)ᵐ	6² ÷ 3² = 2² = 4
a⁰ = 1 (a ≠ 0)	7⁰ = 1
a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8

(−1) raised to an even power is 1; to an odd power is −1.
10⁻² = 1/100 = 0.01.
(2/3)⁻² = (3/2)² = 9/4.
The laws hold only when the bases are the same (for multiplying and dividing powers) or the powers are the same.
2³ + 2³ is 16, but not 2⁶. The laws apply to products, not sums.

Standard form (scientific notation): A number is written as k × 10ⁿ with 1 ≤ k < 10. 5,200 = 5.2 × 10³ and 0.0045 = 4.5 × 10⁻³.

Squares and cubes to remember: 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225, 16² = 256; 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216.

Classroom angle

Introduce algebra as generalised arithmetic: "a number" instead of "5". Use a balance scale to explain equations: if the same weight is removed from both pans, the scale stays level. Use matchstick patterns (such as the number of sticks for n squares) to lead children to expressions like 3n + 1. Use number puzzles ("think of a number…") to show why letters are useful. Let children check each answer by substitution to build confidence.

Exam traps

(a + b)² is not a² + b²; the middle term 2ab is needed.
a² − b² is a difference, while (a − b)² is a square of a difference; they are different.
When removing brackets preceded by a minus sign, change every sign inside.
In 3x + 5 = 20, the 5 moves as −5; do not add it to both sides.
x² and x³ are unlike terms and cannot be combined.
2³ × 2⁴ = 2⁷, not 4⁷: the base stays the same, and the powers are added.
(aᵐ)ⁿ means multiply the powers; aᵐ × aⁿ means add the powers.
Any non-zero number to the power 0 is 1, not 0.

One-liners

Terms are separated by + or −.
Only like terms can be added.
Degree of a polynomial = highest power of the variable.
(a + b)² = a² + 2ab + b².
a² − b² = (a + b)(a − b).
(x + a)(x + b) = x² + (a + b)x + ab.
A linear equation in one variable has one root.
Transposition changes the sign of the term.
aᵐ × aⁿ = aᵐ⁺ⁿ.
(aᵐ)ⁿ = aᵐⁿ.
a⁰ = 1 and a⁻ⁿ = 1/aⁿ.
Standard form is k × 10ⁿ with 1 ≤ k < 10.

Practice questions

The coefficient of x in −7x is
1. x
2. −7
3. 0
4. 7
Answer
B. −7
The number multiplying the variable, with its sign, is the coefficient.
Which pair are like terms?
1. 3x² and 3x
2. 5a and 5b
3. 2xy and 2x
4. 3x² and −7x²
Answer
D. 3x² and −7x²
Like terms have the same variables with the same powers.
An expression with exactly two terms is called a
1. monomial
2. binomial
3. constant
4. trinomial
Answer
B. binomial
Bi means two.
The degree of the polynomial x³ − 2x + 5 is
1. 1
2. 2
3. 5
4. 3
Answer
D. 3
The highest power of x is 3.
The degree of the term x²y³ is
1. 2
2. 3
3. 5
4. 6
Answer
C. 5
Add the powers: 2 + 3 = 5.
(a + b)² equals
1. a² + 2ab + b²
2. a² + ab + b²
3. a² + b²
4. a² − 2ab + b²
Answer
A. a² + 2ab + b²
This is the standard identity for the square of a sum.
a² − b² equals
1. (a + b)²
2. (a + b)(a − b)
3. (a − b)²
4. a(a − b)
Answer
B. (a + b)(a − b)
This is the difference of two squares.
The value of 102² using an identity is
1. 10604
2. 10400
3. 10204
4. 10404
Answer
D. 10404
(100 + 2)² = 10000 + 400 + 4 = 10404.
The value of 103 × 97 is
1. 9971
2. 9991
3. 9999
4. 10009
Answer
B. 9991
(100 + 3)(100 − 3) = 10000 − 9 = 9991.
If a + b = 7 and ab = 10, then a² + b² is
1. 49
2. 19
3. 29
4. 39
Answer
C. 29
a² + b² = (a + b)² − 2ab = 49 − 20 = 29.
If x + 1/x = 5, then x² + 1/x² is
1. 23
2. 27
3. 21
4. 25
Answer
A. 23
Square both sides: x² + 2 + 1/x² = 25, so the sum is 23.
(x + 3)(x + 4) equals
1. x² + 12
2. x² + 7x + 12
3. x² + 7x + 7
4. x² + 12x + 7
Answer
B. x² + 7x + 12
x² + (3 + 4)x + 12.
Solution of 3x + 5 = 20 is
1. 3
2. 15
3. 5
4. 25/3
Answer
C. 5
3x = 15, so x = 5.
Solution of 5x − 7 = 3x + 9 is
1. 8
2. 1
3. 16
4. −8
Answer
A. 8
2x = 16, so x = 8.
Solution of x/3 + 2 = 6 is
1. 4
2. 6
3. 24
4. 12
Answer
D. 12
x/3 = 4, so x = 12.
Solution of (x + 1)/(x + 2) = 3/4 is
1. 1
2. 2
3. 3
4. 5
Answer
B. 2
4x + 4 = 3x + 6, so x = 2.
The sum of three consecutive numbers is 72. The smallest number is
1. 24
2. 22
3. 23
4. 25
Answer
C. 23
3x + 3 = 72 gives x = 23; the numbers are 23, 24, 25.
A rectangle has perimeter 60 cm and its length is twice its breadth. Its breadth is
1. 10 cm
2. 15 cm
3. 20 cm
4. 6 cm
Answer
A. 10 cm
2(2b + b) = 60, so 6b = 60 and b = 10.
A father is 3 times as old as his son and the sum of their ages is 60 years. The son's age is
1. 12 years
2. 20 years
3. 30 years
4. 15 years
Answer
D. 15 years
x + 3x = 60, so x = 15.
The solution of a linear equation in one variable is also called its
1. coefficient
2. exponent
3. degree
4. root
Answer
D. root
The value satisfying the equation is its root.
2³ × 2⁴ equals
1. 2⁷
2. 4⁷
3. 2¹²
4. 4¹²
Answer
A. 2⁷
Same base: add the powers.
5⁶ ÷ 5⁴ equals
1. 25
2. 5
3. 5¹⁰
4. 1
Answer
A. 25
5⁶⁻⁴ = 5² = 25.
(3²)³ equals
1. 27
2. 729
3. 243
4. 81
Answer
B. 729
3² × ³ = 3⁶ = 729.
The value of 7⁰ is
1. 0
2. 1
3. 7
4. undefined
Answer
B. 1
Any non-zero number to the power zero is 1.
The value of 2⁻³ is
1. −8
2. 8
3. 1/8
4. −6
Answer
C. 1/8
2⁻³ = 1/2³ = 1/8.
(2/3)⁻² equals
1. 6/9
2. 4/9
3. 9/4
4. −4/9
Answer
C. 9/4
(2/3)⁻² = (3/2)² = 9/4.
2³ × 5³ equals
1. 10⁶
2. 125
3. 100
4. 1000
Answer
D. 1000
Same power: (2 × 5)³ = 10³ = 1000.
The standard form of 5,200 is
1. 5.2 × 10³
2. 0.52 × 10⁴
3. 52 × 10²
4. 5.2 × 10²
Answer
A. 5.2 × 10³
Standard form has 1 ≤ k < 10.
The number 0.0045 in standard form is
1. 45 × 10⁻⁴
2. 4.5 × 10³
3. 4.5 × 10⁻²
4. 4.5 × 10⁻³
Answer
D. 4.5 × 10⁻³
Move the decimal point 3 places to the right: 4.5 × 10⁻³.
(−1) raised to an odd power equals
1. −1
2. 1
3. 0
4. −2
Answer
A. −1
An odd power of −1 is −1.
If x = 2, the value of 3x² − 5x + 1 is
1. −3
2. 7
3. 13
4. 3
Answer
D. 3
12 − 10 + 1 = 3.
Which method helps children understand equations in Class VI?
1. Memorising solved examples
2. A balance scale with equal weights on both pans
3. Copying the rules of transposition
4. Practising only without checking answers
Answer
B. A balance scale with equal weights on both pans
The balance model shows that both sides must be changed equally.
What should a teacher ask children to do after solving an equation?
1. Leave it unchecked
2. Copy the next problem
3. Substitute the answer back to check it
4. Erase the working
Answer
C. Substitute the answer back to check it
Checking by substitution builds understanding and accuracy.
Statements: 1. (a + b)² = a² + b². 2. (a − b)² = a² − 2ab + b². Which is/are correct?
1. Neither 1 nor 2
2. 1 only
3. 2 only
4. Both 1 and 2
Answer
C. 2 only
The identity for (a + b)² includes the middle term 2ab.
Statements: 1. x² and x³ are like terms. 2. 5x + 3x = 8x. Which is/are correct?
1. Both 1 and 2
2. Neither 1 nor 2
3. 1 only
4. 2 only
Answer
D. 2 only
Like terms need the same power, so x² and x³ are unlike.
Statements: 1. 2³ × 2⁴ = 4⁷. 2. (2³)⁴ = 2¹². Which is/are correct?
1. 2 only
2. Both 1 and 2
3. Neither 1 nor 2
4. 1 only
Answer
A. 2 only
Powers are added when multiplying the same base, so 2³ × 2⁴ = 2⁷.
Statements: 1. Any non-zero number raised to the power 0 equals 1. 2. a⁻ⁿ = 1/aⁿ. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard laws of exponents.
Statements: 1. An identity holds for every value of the variable. 2. An equation holds only for certain values of the variable. Which is/are correct?
1. Neither 1 nor 2
2. 1 only
3. 2 only
4. Both 1 and 2
Answer
D. Both 1 and 2
This is the difference between an identity and an equation.
Statements: 1. A polynomial with three terms is a trinomial. 2. The degree of a non-zero constant is 1. Which is/are correct?
1. Both 1 and 2
2. Neither 1 nor 2
3. 1 only
4. 2 only
Answer
C. 1 only
A non-zero constant has degree 0.
Statements: 1. 2³ + 2³ = 2⁶. 2. 2³ × 2³ = 2⁶. Which is/are correct?
1. 2 only
2. Both 1 and 2
3. Neither 1 nor 2
4. 1 only
Answer
A. 2 only
Laws of exponents apply to products; 2³ + 2³ = 16 = 2⁴.
Match the expression with its expansion: P. (a + b)² Q. (a − b)² R. a² − b² S. (x + a)(x + b) 1. a² − 2ab + b² 2. (a + b)(a − b) 3. a² + 2ab + b² 4. x² + (a + b)x + ab
1. P-3, Q-2, R-1, S-4
2. P-3, Q-1, R-2, S-4
3. P-3, Q-1, R-4, S-2
4. P-1, Q-3, R-2, S-4
Answer
B. P-3, Q-1, R-2, S-4
These are the standard identities.
Match the law with the example: P. aᵐ × aⁿ Q. aᵐ ÷ aⁿ R. (aᵐ)ⁿ S. a⁰ 1. 5⁶ ÷ 5⁴ = 5² 2. 2³ × 2⁴ = 2⁷ 3. 7⁰ = 1 4. (3²)³ = 3⁶
1. P-4, Q-1, R-2, S-3
2. P-2, Q-1, R-4, S-3
3. P-2, Q-1, R-3, S-4
4. P-1, Q-2, R-4, S-3
Answer
B. P-2, Q-1, R-4, S-3
Add powers to multiply, subtract to divide, multiply powers for a power of a power.
If the sum of two numbers is 18 and their difference is 4, the numbers are
1. 10 and 8
2. 11 and 7
3. 12 and 6
4. 13 and 5
Answer
B. 11 and 7
x + y = 18 and x − y = 4 give x = 11, y = 7.
The number 10⁻² equals
1. 0.01
2. 0.1
3. 0.001
4. −100
Answer
A. 0.01
10⁻² = 1/100 = 0.01.
Simplify: (5a − 3) − (2a − 4)
1. 3a − 7
2. 3a − 1
3. 3a + 1
4. 7a + 1
Answer
C. 3a + 1
Change the signs of the second bracket: 5a − 3 − 2a + 4 = 3a + 1.

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1.The difference between a number and the square root of the number is 2. The number is:
2.The difference between two numbers is 8. If twice the larger number is 4 more than three times the smaller number, what is the larger number?
3.The expression x4 – 8x2 + m will be a perfect square when the value of m is:
4.The expression √(10 + 2(√6 - √15 - √10)) is equal to:
5.Seven years ago Ramu was five times as old as his son. The present age difference between Ramu and his son is 28 years. Find the sum of their present ages.
6.The ages of two siblings add up to 44 years. If one sibling is 4 years younger than twice the other's age, what is the difference between their ages?
7.The ages of a mother and daughter are 36 years and 12 years, respectively. In how many years will the mother be twice the age of the daughter?
8.Simplify the expression (36p² + 49q²) (6p + 7q) (6p – 7q)
9.Simplify (2pq + 11q)2 - (2pq - 11q)2.
10.The sum of two positive numbers is 26 and their product is 144. The positive difference between them is:

10 left. You can check any time.