Algebraic expressions are mathematical phrases that combine numbers, variables, and operations (like addition, subtraction, multiplication, and division).
Key concepts:
- Variables: Letters or symbols that represent unknown values (e.g., x, y, a, b).
- Constants: Fixed numerical values (e.g., 2, -5, 7).
- Terms: Parts of an algebraic expression separated by addition or subtraction (e.g., 3x, -2y, 5).
- Coefficients: The numerical factor of a term (e.g., 3 in 3x, -2 in -2y).
- Like terms: Terms with the same variable raised to the same power (e.g., 2x and 5x).
- Unlike terms: Terms with different variables or the same variable raised to different powers (e.g., 2x and 3y, 4x² and 2x).
Operations on algebraic expressions:
- Addition and subtraction: Combine like terms.
- Multiplication: Multiply coefficients and variables according to the rules of exponents.
- Division: Divide coefficients and simplify variables using exponent rules.
Evaluating algebraic expressions:
- Substitute given values for variables and simplify the expression using arithmetic operations.
Example:
- The expression 3x + 2y – 5 has three terms: 3x, 2y, and -5.
- The coefficients are 3 and 2.
- If x = 2 and y = 3, the value of the expression is 3(2) + 2(3) – 5 = 7.
By understanding these concepts, you can build a strong foundation for further algebraic studies.
Exercise 10.1
1. Get the algebraic expressions in the following cases using variables, constants and arithmetic operations:
(i) Subtraction of z from y.
(ii) One half of the sum of numbers x and y.
(iii) The number z multiplied by itself.
(iv) One-fourth of the product of numbers p and q.
(v) Numbers x and y both squared and added.
(vi) Number 5 added to three times the product of number m and n.
(vii) Product of numbers y and 2 subtracted from 10.
(viii) Sum of numbers a and b subtracted from their product.
Ans :
(i)
Algebraic expression: y – z
(ii)
Algebraic expression: (x + y)/2
(iii)
Algebraic expression: z * z or z²
(iv)
Algebraic expression: (p * q) / 4
(v)
Algebraic expression: x² + y²
(vi)
Algebraic expression: 3mn + 5
(vii)
Algebraic expression: 10 – 2y
(viii)
Algebraic expression: ab – (a + b)
2.
(i) Identify the terms and their factors in the following expressions show the terms and factors by tree diagrams.
(a) x – 3
(b) 1 + x + x2
(c) y – y3
(d) 5xy2 + 7x2y
(e) -ab + 2b2 – 3a2
Ans :
(ii) Identify terms and factors in the expression
given below:
(a) -4x + 5
(b) -4x + 5y
(c) 5y + 3y2
(d) xy + 2x2y2
(e) pq + q
(f) 1.2ab – 2.4b + 3.6a
(g) ¾*x+1/4
(h) 0.1p2 + 0.2q2
Ans :
Analyzing the Expressions
(a) -4x + 5
- Terms: -4x, 5
- Factors:
- -4x: -4, x
- 5: 5
(b) -4x + 5y
- Terms: -4x, 5y
- Factors:
- -4x: -4, x
- 5y: 5, y
(c) 5y + 3y²
- Terms: 5y, 3y²
- Factors:
- 5y: 5, y
- 3y²: 3, y, y
(d) xy + 2x²y²
- Terms: xy, 2x²y²
- Factors:
- xy: x, y
- 2x²y²: 2, x, x, y, y
(e) pq + q
- Terms: pq, q
- Factors:
- pq: p, q
- q: q
(f) 1.2ab – 2.4b + 3.6a
- Terms: 1.2ab, -2.4b, 3.6a
- Factors:
- 1.2ab: 1.2, a, b
- -2.4b: -2.4, b
- 3.6a: 3.6, a
(g) 3/4x + 1/4
- Terms: 3/4x, 1/4
- Factors:
- 3/4x: 3/4, x
- 1/4: 1/4
(h) 0.1p² + 0.2q²
- Terms: 0.1p², 0.2q²
- Factors:
- 0.1p²: 0.1, p, p
- 0.2q²: 0.2, q, q
3. Identify the numerical coefficients of terms (other than constants) in the following:
(i) 5 – 3t2
(ii) 1 + t + t2 + t3
(iii) x + 2xy + 3y
(iv) 100m + 1000n
(v) -p2q2 + 7pq
(vi) 1.2 a + 0.86
(vii) 3.14r2
(viii) 2(l + b)
(ix) 0.1y + 0.01y2
Ans :
i) 5 – 3t²
Numerical coefficient: -3
ii) 1 + t + t² + t³
Numerical coefficients: 1, 1, 1
iii) x + 2xy + 3y
Numerical coefficients: 2, 3
iv) 100m + 1000n
Numerical coefficients: 100, 1000
v) -p²q² + 7pq
Numerical coefficients: -1, 7
vi) 1.2a + 0.86
Numerical coefficient: 1.2
vii) 3.14r²
Numerical coefficient: 3.14
viii) 2(l + b)
Numerical coefficient: 2 (when expanded, becomes 2l + 2b)
ix) 0.1y + 0.01y²
Numerical coefficients: 0.1, 0.01
4 (a) Identify terms which contain x and give the
coefficient of x.
(i) y2x + y
(ii) 13y2 – 8yx
(iii) x + y + 2
(iv) 5 + z + zx
(v) 1 + x + xy
(vi) 12 xy2 + 25
(vii) 7x + xy2
Ans :
Analyzing the Expressions
(i) y²x + y
- Term containing x: y²x
- Coefficient of x: y²
(ii) 13y² – 8yx
- Term containing x: -8yx
- Coefficient of x: -8y
(iii) x + y + 2
- Term containing x: x
- Coefficient of x: 1 (implied)
(iv) 5 + z + zx
- Terms containing x: x, zx
- Coefficient of x in x: 1
- Coefficient of x in zx: z
(v) 1 + x + xy
- Terms containing x: x, xy
- Coefficient of x in x: 1
- Coefficient of x in xy: y
(vi) 12xy² + 25
- Term containing x: 12xy²
- Coefficient of x: 12y²
(vii) 7x + xy²
- Terms containing x: 7x, xy²
- Coefficient of x in 7x: 7
- Coefficient of x in xy²: y²
(b) Identify terms which contain y2 and give the coefficients of y2.
(i) 8 – xy2
(ii) 5y2 + 7x
(iii) 2x2y – 15xy2 + 7y2
Ans :
Analyzing the Expressions
(i) 8 – xy²
- Term containing y²: -xy²
- Coefficient of y²: -x
(ii) 5y² + 7x
- Term containing y²: 5y²
- Coefficient of y²: 5
(iii) 2x²y – 15xy² + 7y²
- Terms containing y²: -15xy², 7y²
- Coefficient of y² in -15xy²: -15x
- Coefficient of y² in 7y²: 7
5. Classify into monomials, binomials and trinomials:
(i) 4y – 7x
(ii) y2
(iii) x + y – xy
(iv) 100
(v) ab – a – b
(vi) 5 – 3t
(vii) 4p2q – 4pq2
(viii) 7mn
(ix) z2 – 3z + 8
(x) a2 + b2
(xi) z2 + z
(xii) 1 + x + x2
Ans :
| Expression | Type |
| (i) 4y – 7x | Binomial |
| (ii) y² | Monomial |
| (iii) x + y – xy | Trinomial |
| (iv) 100 | Monomial |
| (v) ab – a – b | Trinomial |
| (vi) 5 – 3t | Binomial |
| (vii) 4p²q – 4pq² | Binomial |
| (viii) 7mn | Monomial |
| (ix) z² – 3z + 8 | Trinomial |
| (x) a² + b² | Binomial |
| (xi) z² + z | Binomial |
| (xii) 1 + x + x² | Trinomial |
6. State whether a given pair of terms is of like or unlike terms.
(i) 1, 100
(ii) -7x, 5/2*x
(iii) -29x, -29y
(iv) 14xy, 42yx
(v) 4m2p, 4mp2
(vi) 12xz, 12 x2y2
Ans :
Analysis
(i) 1, 100
- Both terms are constants, so they are like terms.
(ii) -7x, (5/2)x
- Both terms have the same variable (x) with the same power (1), so they are like terms.
(iii) -29x, -29y
- The terms have different variables (x and y), so they are unlike terms.
(iv) 14xy, 42yx
- Both terms have the same variables (x and y) with the same powers, so they are like terms.
(v) 4m²p, 4mp²
- The variables and their powers are different, so they are unlike terms.
(vi) 12xz, 12x²z²
- The variables and their powers are different, so they are unlike terms.
7. Identify like terms in the following:
(a)-xy2, -4yx2, 8x2, 2xy2, 7y2, -11x2, -100x, -11yx, 20x2y, -6x2, y, 2xy, 3x
(b) 10pq, 7p, 8q, -p2q2, -7qp, -100q, -23, 12q2p2, -5p2, 41, 2405p, 78qp, 13p2q, qp2, 701p2
Ans :
(a) -xy², -4yx², 8x², 2xy², 7y², -11x², -100x, -11yx, 20x²y, -6x², y, 2xy, 3x
- Like terms with x²: 8x², -11x², -6x², 20x²y
- Like terms with xy: -xy², 2xy², -11yx, 2xy
- Like terms with x: -100x, 3x
- Like terms with y²: 7y²
- Like terms with y: y
(b) 10pq, 7p, 8q, -p²q², -7qp, -100q, -23, 12q²p², -5p², 41, 2405p, 78qp, 13p²q, qp², 701p²
- Like terms with pq: 10pq, -7qp, 78qp
- Like terms with p²q²: -p²q², 12q²p², 13p²q, qp²
- Like terms with p²: -5p², 701p²
- Like terms with p: 7p, 2405p
- Like terms with q: 8q, -100q
- Like terms with constants: -23, 41
Exercise 10.2
1. If m = 2, find the value of:
(i) m – 2
(ii) 3m – 5
(iii) 9 – 5m
(iv) 3m2 – 2m – 7
(v) 5m/2−4
Ans :
Let’s substitute m = 2 into the expressions:
(i) m – 2
= 2 – 2 = 0
(ii) 3m – 5
= 3(2) – 5 = 6 – 5 = 1
(iii) 9 – 5m
= 9 – 5(2) = 9 – 10 = -1
(iv) 3m² – 2m – 7
= 3(2)² – 2(2) – 7 = 3(4) – 4 – 7 = 12 – 4 – 7 = 1
(v) (5m)/2 – 4
= (5 * 2) / 2 – 4 = 10 / 2 – 4 = 5 – 4 = 1
2. If p = -2, find the value of:
(i) 4p + 7
(ii) -3p2 + 4p + 7
(iii) -2p3 – 3p2 + 4p + 7
Ans :
Substituting p = -2
(i) 4p + 7
= 4(-2) + 7 = -8 + 7 = -1
(ii) -3p² + 4p + 7
= -3(-2)² + 4(-2) + 7 = -3(4) – 8 + 7 = -12 – 8 + 7 = -13
(iii) -2p³ – 3p² + 4p + 7
= -2(-2)³ – 3(-2)² + 4(-2) + 7 = -2(-8) – 3(4) – 8 + 7 = 16 – 12 – 8 + 7 = 3
3. Find the value of the following expressions, when x = –1:
(i) 2x – 7 (ii) – x + 2 (iii) x 2 + 2x + 1 (iv) 2x 2 – x – 2
Ans :
(i) 2x – 7
= 2(-1) – 7 = -2 – 7 = -9
(ii) -x + 2
= -(-1) + 2 = 1 + 2 = 3
(iii) x² + 2x + 1
= (-1)² + 2(-1) + 1 = 1 – 2 + 1 = 0
4. If a = 2, b = -2, find the value of:
(i) a2 + b2
(ii) a2 + ab + b2
(iii) a2 – b2
Ans :
Given:
- a = 2
- b = -2
Calculations:
(i) a² + b² = (2)² + (-2)² = 4 + 4 = 8
(ii) a² + ab + b² = (2)² + (2)(-2) + (-2)² = 4 – 4 + 4 = 4
(iii) a² – b² = (2)² – (-2)² = 4 – 4 = 0
5. When a = 0, b = -1, find the value of the given expressions:
(i) 2a + 2b
(ii) 2a2 + b2 + 1
(iii) 2a2b + 2ab2 + ab
(iv) a2 + ab + 2
Ans :
Given:
- a = 0
- b = -1
Calculations:
(i) 2a + 2b = 2(0) + 2(-1) = 0 – 2 = -2
(ii) 2a² + b² + 1 = 2(0)² + (-1)² + 1 = 0 + 1 + 1 = 2
(iii) 2a²b + 2ab² + ab = 2(0)²(-1) + 2(0)(-1)² + (0)(-1) = 0 + 0 + 0 = 0
(iv) a² + ab + 2 = (0)² + (0)(-1) + 2 = 0 + 0 + 2 = 2
6. Simplify the expressions and find the value if x is equal to 2.
(i) x + 7 +4(x – 5)
(ii) 3(x + 2) + 5x – 7
(iii) 6x + 5(x – 2)
(iv) 4(2x – 1) + 3x + 11
Ans :
(i) x + 7 + 4(x – 5)
- Simplify the expression within the parentheses:
- x + 7 + 4x – 20
- Combine like terms:
- 5x – 13
- Substitute x = 2:
- 5(2) – 13 = 10 – 13 = -3
(ii) 3(x + 2) + 5x – 7
- Simplify the expression within the parentheses:
- 3x + 6 + 5x – 7
- Combine like terms:
- 8x – 1
- Substitute x = 2:
- 8(2) – 1 = 16 – 1 = 15
(iii) 6x + 5(x – 2)
- Simplify the expression within the parentheses:
- 6x + 5x – 10
- Combine like terms:
- 11x – 10
- Substitute x = 2:
- 11(2) – 10 = 22 – 10 = 12
(iv) 4(2x – 1) + 3x + 11
- Simplify the expression within the parentheses:
- 8x – 4 + 3x + 11
- Combine like terms:
- 11x + 7
- Substitute x = 2:
- 11(2) + 7 = 22 + 7 = 29
7. Simplify these expressions and find their values if x = 3, a = -1, b = -2.
(i) 3x – 5 – x + 9
(ii) 2 – 8x + 4x + 4
(iii) 3a + 5 – 8a + 1
(iv) 10 – 3b – 4 – 55
(v) 2a – 2b – 4 – 5 + a
Ans :
(i) 3x – 5 – x + 9
Combine like terms:
- 2x + 4 Substitute x = 3:
- 2(3) + 4 = 6 + 4 = 10
(ii) 2 – 8x + 4x + 4
Combine like terms:
- -4x + 6 Substitute x = 3:
- -4(3) + 6 = -12 + 6 = -6
(iii) 3a + 5 – 8a + 1
Combine like terms:
- -5a + 6 Substitute a = -1:
- -5(-1) + 6 = 5 + 6 = 11
(iv) 10 – 3b – 4 – 5
Combine like terms:
- -3b + 1 Substitute b = -2:
- -3(-2) + 1 = 6 + 1 = 7
(v) 2a – 2b – 4 – 5 + a
Combine like terms:
- 3a – 2b – 9 Substitute a = -1 and b = -2:
- 3(-1) – 2(-2) – 9 = -3 + 4 – 9 = -8
8. (i) If z = 10, find the value of z2 – 3(z – 10).
(ii) If p = -10, find the value of p2 -2p – 100.
Ans :
Let’s substitute the given values and solve the expressions.
(i) If z = 10, find the value of z² – 3(z – 10)
- Substitute z with 10:
- 10² – 3(10 – 10)
- 100 – 3(0)
- 100 – 0
- 100
(ii) If p = -10
p² – 2p – 100
- Substitute p with -10:
- (-10)² – 2(-10) – 100
- 100 + 20 – 100
- 20
9. What should be the value of a if the value of 2x2 + x – a equals to 5, when x = 0?
Ans :
We have the expression: 2x² + x – a
The value of the expression is 5 when x = 0.
We need to find the value of ‘a’.
Solution:
Substitute x = 0 into the expression:
- 2(0)² + (0) – a = 5
- 0 + 0 – a = 5
- -a = 5
To find the value of ‘a’, we multiply both sides by -1:
- a = -5
Therefore, the value of ‘a’ is -5.
10. Simplify the expression and find its value when a = 5 and b = -3.
2(a2 + ab) + 3 – ab
Ans :
Let’s simplify the expression and find its value.
Simplifying the expression
- 2(a² + ab) + 3 – ab
- Distribute the 2:
- 2a² + 2ab + 3 – ab
- Combine like terms:
- 2a² + ab + 3
Finding the value when a = 5 and b = -3
Substitute a = 5 and b = -3 into the simplified expression:
- 2(5)² + (5)(-3) + 3
- 2(25) – 15 + 3
- 50 – 15 + 3
- 38
Therefore, the simplified expression is 2a² + ab + 3, and its value when a = 5 and b = -3 is 38.
