Concept 1
Read the parts of an expression
In 5x²−3x+7, the terms are 5x², −3x and 7; the coefficients are 5 and −3; the constant is 7. Only like terms—terms with identical variable parts—can be combined.
Substitution: replace each variable with its value, use brackets for negative values, and follow the order of operations.
Concept 2
Use identities as reusable patterns
| Identity | Expansion |
|---|---|
| (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 |
Recognise the complete pattern before applying it. For example, 99²=(100−1)²=10,000−200+1=9,801.
Concept 3
Factorisation reverses expansion
First remove the greatest common factor. Then look for an identity or split the middle term. Thus 6x²+9x=3x(2x+3), while x²−25=(x−5)(x+5).
For x²+7x+12, find two numbers with sum 7 and product 12: 3 and 4. Therefore x²+7x+12=(x+3)(x+4).
Concept 4
Keep both sides balanced
Perform the same operation on both sides. Solve 3(x−2)+5=2x+9: expand to 3x−1=2x+9, subtract 2x, then add 1, giving x=10.
Clear fractions by multiplying every term by the least common denominator. Check the result in the original equation, especially when a denominator contains the variable.
Concept 5
Locate points with ordered pairs
A point (x,y) moves x units horizontally and y units vertically. Quadrant signs are I (+,+), II (−,+), III (−,−), IV (+,−). Points on the x-axis have y=0; points on the y-axis have x=0.
Distance along a horizontal or vertical line is the absolute difference of the changing coordinates. The midpoint of (x₁,y₁) and (x₂,y₂) is ((x₁+x₂)/2,(y₁+y₂)/2).
Concept 6
A linear equation draws a straight line
Write y=mx+c: m is slope and c is the y-intercept. Slope m=(change in y)/(change in x). Positive slope rises left to right; negative slope falls; zero slope is horizontal.
For 2x+y=6, y=−2x+6. Two convenient points are (0,6) and (3,0). Plotting any two correct points determines the line.
Concept 7
The intersection solves two equations
For two linear equations, an intersecting pair has one solution, distinct parallel lines have none, and the same line has infinitely many. Algebraic substitution or elimination must agree with the graphical intersection.
For x+y=7 and x−y=1, adding gives 2x=8, so x=4 and y=3. The two graph lines meet at (4,3).
Worked methods
Examples with reasoning
Simplify
4x−3+2x+8=6x+5 by combining like terms.
Identity
103×97=(100+3)(100−3)=10000−9=9991.
Factorise
2x²−8=2(x²−4)=2(x−2)(x+2).
Solve an equation
5x−7=3x+9 gives 2x=16, so x=8.
Read a line
y=3x−2 has slope 3 and y-intercept −2; when x=2, y=4.
Find an intersection
x+y=9 and x−y=3 add to 2x=12, so their intersection is (6,3).
Exam traps
Errors that silently change the problem
Combining unlike terms
3x and 3x² are not like terms.
Missing the middle term
(a+b)² includes 2ab, not only a²+b².
Losing a minus sign
Use brackets before distributing a negative multiplier.
Operating on one side
An equation stays equivalent only when both sides remain balanced.
Reversing coordinates
(x,y) means horizontal first, vertical second.
Reading intercept as slope
In y=mx+c, m is slope and c is the y-intercept.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute sheet
- Combine only terms with the same variable part.
- (a±b)²=a²±2ab+b²; (a+b)(a−b)=a²−b².
- Factorisation reverses multiplication and expansion.
- Apply each equation operation to both sides.
- Coordinates are ordered (x,y): horizontal, then vertical.
- In y=mx+c, m is slope and c is the y-intercept.
- The intersection of two lines is their common solution.
Source ledger
Official foundations
- SSC CGL official syllabus: Algebra and Graphs of Linear Equations
- NCERT Mathematics contents: polynomials and linear equations
- NCERT official exemplar directory
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Drafted 25 September 2026.
Geometry.