4.1 Algebra & Linear Equations
Key Takeaways
- RRB Group D algebra rewards translation: read the word problem, assign one variable for the unknown, and write one equation before solving.
- The four core identities (a+b)^2 = a^2+2ab+b^2, (a-b)^2 = a^2-2ab+b^2, a^2-b^2 = (a+b)(a-b), and (x+a)(x+b) = x^2+(a+b)x+ab appear repeatedly; memorise them cold.
- For a pair of linear equations in two variables, the elimination method is fastest when coefficients align, and substitution is cleaner when one variable is already isolated.
- A quadratic x^2+bx+c with factors (x+p)(x+q) needs p+q = b and pq = c; pick factor pairs of c whose signed sum equals b.
- With 1/3 negative marking, never guess a 50/50 algebra question you cannot reduce to two options — one wrong answer costs 0.33 marks.
Why Algebra Matters for RRB Group D
The Mathematics section of the RRB Group D CBT carries 25 questions out of 100, and algebra-flavoured problems — simplifications, identities, linear-equation word problems, and the occasional quadratic — routinely account for 8 to 12 of those 25 marks. Algebra is also the silent engine behind Age, Clock, and many Data Interpretation problems, so the time you spend here pays dividends across the paper.
Algebraic Expressions and Simplification
An algebraic expression combines constants and variables with the operations +, −, ×, ÷. A term such as 7x has coefficient 7 and variable x. Like terms share the same variable raised to the same power, so 5x and -2x combine to 3x, but 5x and 5x^2 do not.
Simplification rules used in RRB questions:
- Distribute the sign across a bracket:
a - (b - c) = a - b + c. The single most common error is dropping the minus sign before a bracket. - Combine like terms only.
- Respect order of operations: brackets, powers, multiplication/division (left to right), addition/subtraction.
Polynomials and Standard Identities
A polynomial has non-negative integer powers of the variable. By degree: constant (degree 0), linear (1), quadratic (2), cubic (3). RRB rarely goes beyond quadratics.
Four identities you must internalise:
| Identity | Expanded Form | Quick Use |
|---|---|---|
| (a+b)^2 | a^2 + 2ab + b^2 | Square of a sum |
| (a-b)^2 | a^2 - 2ab + b^2 | Square of a difference |
| a^2 - b^2 | (a+b)(a-b) | Difference of squares — the most tested identity |
| (x+a)(x+b) | x^2 + (a+b)x + ab | Factorising quadratics with leading coefficient 1 |
A fifth pair worth knowing for faster multiplication: (a+b)^2 - (a-b)^2 = 4ab and (a+b)^2 + (a-b)^2 = 2(a^2+b^2).
Linear Equations in One Variable
Solve by isolating the variable. The golden rule: whatever you do to one side, do to the other. Transpose terms by changing their sign.
Worked example 1. Solve 3(2x - 5) + 2 = 2x - (x - 7).
Step 1 — expand: 6x - 15 + 2 = 2x - x + 7.
Step 2 — simplify both sides: 6x - 13 = x + 7.
Step 3 — collect x-terms left, constants right: 6x - x = 7 + 13 → 5x = 20.
Step 4 — divide: x = 4.
Check: LHS 3(8-5)+2 = 11; RHS 8-(4-7)=8-(-3)=11. Verified.
Linear Equations in Two Variables
A pair such as 2x + 3y = 16 and x - y = 1 can be solved by substitution or elimination.
Worked example 2 (elimination). Solve 2x + 3y = 16 and x + y = 6.
Multiply the second equation by 3 so the y-coefficients match: 3x + 3y = 18. Now subtract the first equation from it: (3x + 3y) - (2x + 3y) = 18 - 16 → x = 2. Back-substitute into x + y = 6: y = 4.
Check both equations: 2(2) + 3(4) = 4 + 12 = 16 ✓ and 2 + 4 = 6 ✓.
Worked example 2b (substitution). The same pair by the other method: from x + y = 6, write x = 6 - y. Substitute into the first: 2(6 - y) + 3y = 16 → 12 - 2y + 3y = 16 → y = 4, so x = 2. Same answer, as it must be.
Choose the method by inspection: elimination when a variable's coefficients can be matched in one multiplication, substitution when one variable is already isolated (as x + y = 6 nearly is). Always back-substitute into both original equations before marking — a sign slip usually survives one check but not two.
Quadratic Basics by Factorisation
For x^2 + bx + c, find two numbers p and q with p + q = b and p × q = c. Then x^2 + bx + c = (x + p)(x + q). Watch signs carefully:
- c positive, b positive → both factors positive.
- c positive, b negative → both factors negative.
- c negative → one positive, one negative; the larger absolute value follows the sign of b.
Worked example 3. Factorise x^2 - 5x + 6.
Need p + q = -5 and p × q = 6. The pair (-2, -3) works: sum -5, product 6. So x^2 - 5x + 6 = (x - 2)(x - 3). Roots: x = 2 and x = 3.
Translating Word Problems into Equations
The RRB algebra question is usually a short story. Translate phrase by phrase:
| English phrase | Algebra |
|---|---|
| "a number" | let it be x |
| "twice the number" | 2x |
| "5 more than the number" | x + 5 |
| "3 less than twice the number" | 2x - 3 |
| "sum is 20" | (expression) = 20 |
| "three consecutive integers" | x, x+1, x+2 |
| "two-digit number, tens digit a, units b" | 10a + b |
Worked example 4. The sum of two numbers is 35 and their difference is 5. Find the smaller.
Let the numbers be x and y with x > y. Then x + y = 35 and x - y = 5. Add: 2x = 40 → x = 20, so y = 15. The smaller number is 15. Notice the shortcut: the larger is (sum + difference)/2 = 20 and the smaller is (sum - difference)/2 = 15.
Exam Traps and Time Strategy
- Sign errors in expansion of
(a - b)^2— the middle term is-2ab, not+2ab. - Forgetting negative marking: a guess with two plausible options and two you cannot eliminate is a net loss in expectation; skip instead.
- Reading "less than" backwards: "3 less than x" is
x - 3, NOT3 - x. - Units digit problems: form
10t + ucorrectly; swapping digits gives10u + t. - Aim to spend at most 50 seconds on an algebra simplification and 70 seconds on a two-variable word problem; flag longer ones for review.
Quick Practice
Reduce each problem to an equation before calculating. If your equation is right, the arithmetic is usually trivial — that is the whole point of algebra on this exam.
If 3 times a number added to 7 gives the same result as 5 times the number minus 9, what is the number?
The sum of two numbers is 48 and one is four times the other. What is the smaller number?
Factorise x^2 + 7x + 12 completely.