Pariksha SaathiLesson 3 of 8 · Mathematics, Classes VI-VIII (Paper II)
Algebra: Expressions, Linear Equations, Identities and Factorisation
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Variables and expressions
A variable is a letter that can take different values, while a constant has a fixed value. Class VI algebra starts from patterns: n letters L made of matchsticks need 2n sticks and n letters C need 3n. In the expression 5xy² − 3x²y + 7 the terms are 5xy², −3x²y and 7, the numerical coefficient of xy² is 5 and 7 is the constant term; expressions of one, two and three terms are monomials, binomials and trinomials. Like terms have the same variables raised to the same powers (7ab and −2ba are like; 3x²y and 3xy² are not), and only like terms can be added. Translating words needs care: '5 less than twice a number' is 2x − 5, not 5 − 2x.
Working with expressions
Subtracting an expression changes the sign of each of its terms: (7x² − 4x + 2) − (3x² + x − 6) = 4x² − 5x + 8. In multiplication the coefficients multiply and powers of the same variable add: (−3x²y) × (4xy³) = −12x³y⁴. Each term of one bracket multiplies each term of the other: (2x + 3)(x − 5) = 2x² − 10x + 3x − 15 = 2x² − 7x − 15, which can be checked by putting x = 1 (5 × (−4) = −20 and 2 − 7 − 15 = −20).
Linear equations in one variable
A linear equation has its variable only to the power 1, and its solution is the value that makes both sides equal. Whatever is done to one side is done to the other; transposing a term with its sign changed is a short form of this. With the variable on both sides, collect it on one side: 5x − 3 = 3x + 7 gives 2x = 10 and x = 5. With fractions, multiply through by the LCM of the denominators: x/2 − 1/5 = x/3 + 1/4 times 60 becomes 30x − 12 = 20x + 15, so 10x = 27 and x = 27/10, when both sides equal 23/20. Likewise (6x + 1)/3 + 1 = (x − 3)/6 times 6 gives 12x + 2 + 6 = x − 3, so 11x = −11 and x = −1.
Word problems
Name the unknown, write the equation the words describe, solve it and check the answer against the story. The sum of three consecutive multiples of 8 is 888: with 8x, 8(x + 1) and 8(x + 2), 24x + 24 = 888, so x = 36 and the numbers are 288, 296 and 304. A two-digit number has digits adding up to 9 and grows by 27 when its digits are interchanged; with units digit b it is 10(9 − b) + b, the condition gives 18b − 81 = 27, so b = 6 and the number is 36.
Algebraic identities
An identity is true for every value of its variables, an equation only for particular values. The class VIII identities are (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b² and (x + a)(x + b) = x² + (a + b)x + ab. They make hard arithmetic easy: 103² = 10,000 + 600 + 9 = 10,609; 99² = 10,000 − 200 + 1 = 9,801; 197 × 203 = 200² − 3² = 39,991; and 51² − 49² = 100 × 2 = 200. The commonest error, (a + b)² = a² + b², fails for a = 1 and b = 2 (9 against 5); a square of side a + b shows the two missing rectangles of area ab.
Factorisation
Factorising writes an expression as a product, the reverse of expanding. Take out the common factor: 12a²b + 15ab² = 3ab(4a + 5b). Regroup when no factor is common to all terms: 2xy + 2y + 3x + 3 = 2y(x + 1) + 3(x + 1) = (x + 1)(2y + 3). Recognise identities: x² + 8x + 16 = (x + 4)², 4y² − 12y + 9 = (2y − 3)² and 49p² − 36 = (7p − 6)(7p + 6). For x² + px + q, find two numbers with product q and sum p: x² + 5x + 6 = (x + 2)(x + 3), y² − 7y + 12 = (y − 3)(y − 4) and z² − 4z − 12 = (z − 6)(z + 2).
In the classroom
Pupils who read 3a as 'three apples' treat the letter as an object rather than a number, and then write 3x + 5x = 8x², 3 + 2x = 5x or (2a)² = 2a². The teacher asks them to find and correct such errors (x + 2x + 3x is 6x, not 5x), to test any rule by substituting a number, and uses algebra tiles or area diagrams for products and identities.
Key facts
- Like terms: 7ab and −2ba are like terms; 3x²y and 3xy² are unlike, so 3x²y + 3xy² cannot be simplified.
- '5 less than twice a number' is 2x − 5; 'a number decreased by 5 and then doubled' is 2(x − 5).
- (7x² − 4x + 2) − (3x² + x − 6) = 4x² − 5x + 8, and (−3x²y) × (4xy³) = −12x³y⁴.
- 5x − 3 = 3x + 7 gives x = 5; x/2 − 1/5 = x/3 + 1/4 gives x = 27/10 after multiplying both sides by 60.
- Three consecutive multiples of 8 that add up to 888 are 288, 296 and 304.
- Identities: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b²; (x + a)(x + b) = x² + (a + b)x + ab.
- Mental arithmetic with identities: 103² = 10,609; 99² = 9,801; 197 × 203 = 39,991; 51² − 49² = 200.
- Factorisation: 2xy + 2y + 3x + 3 = (x + 1)(2y + 3); 49p² − 36 = (7p − 6)(7p + 6); z² − 4z − 12 = (z − 6)(z + 2).
- Check by substitution: for a = 1 and b = 2, (a + b)² = 9 but a² + b² = 5, so (a + b)² is not a² + b².
Quick revision
Variable takes different values; terms, coefficients, like terms; only like terms add; subtracting an expression changes every sign; linear equation: same operation on both sides, clear fractions with the LCM, check by substitution; identity true for all values; (a ± b)² = a² ± 2ab + b², (a + b)(a − b) = a² − b², (x + a)(x + b) = x² + (a + b)x + ab; factorise by common factor, regrouping, identities or two numbers with product q and sum p.