A comprehensive mathematics guide to algebraic language, manipulation, equations, inequalities, graphs, sequences, functions and advanced problem-solving.
Algebra is the language of patterns, relationships and unknown quantities. At first, it may look like ordinary arithmetic with letters added to it, but its real purpose is much more powerful: algebra allows you to describe a general rule, solve for an unknown, model a real situation, and reason about quantities that can change. For an student, strong algebra is not one isolated chapter. It is the foundation underneath equations, graphs, sequences, functions, coordinate geometry, variation, indices, surds and many parts of problem-solving.
Arithmetic usually asks you to calculate with known numbers. Algebra introduces variables so that one statement can represent many possible values. In 3x + 5, the letter x represents a number whose value may be known later, restricted by a condition, or allowed to vary.
A constant is a fixed number, while a coefficient is the numerical factor multiplying a variable. An algebraic expression is a combination of numbers, variables and operations, but it does not contain an equals sign as part of the expression itself. An equation states that two expressions are equal. An inequality compares expressions using symbols such as <, >, ≤ and ≥.
| Algebraic idea | Example | What it means |
|---|---|---|
| Variable | x | A quantity whose value can vary |
| Constant | 7 | A fixed numerical value |
| Coefficient | 5 in 5x | The number multiplying the variable |
| Term | −3x² | A single part separated by + or − |
| Expression | 2x² + 3x − 1 | A combination of terms |
| Equation | 2x + 1 = 9 | A statement that two quantities are equal |
| Inequality | 2x + 1 > 9 | A comparison giving a range of possible values |
“Five more than twice a number n” becomes 2n + 5.
“Seven less than three times x” becomes 3x − 7.
“The product of x and the number immediately after x” becomes x(x + 1).
Be careful with order: “five less than x” is x − 5, not 5 − x.
Correct notation prevents many mathematics mistakes. Multiplication is normally written without a multiplication sign: 3 × x = 3x. The expression xy means x multiplied by y. A square means a number multiplied by itself, so x² = x × x. The expression 2x² means 2 × x², not (2x)².
Substitution means replacing a variable with a known value. Substitute the entire value, especially when it is negative or fractional, using brackets first.
If x = −3, evaluate 2x² − 5x + 1.
2(−3)² − 5(−3) + 1 = 2(9) + 15 + 1 = 34.
The brackets matter: (−3)² = 9, whereas −3² is interpreted as −9 under standard order of operations.
Substitution is also used in formulas. If the area of a triangle is A = ½bh, then b = 8 and h = 5 give A = ½(8)(5) = 20 square units. Always identify the unit when the formula represents a physical quantity.
Do not treat substitution as simply “putting numbers in”. First identify the variable, copy the formula accurately, substitute with brackets where necessary, simplify step by step, and check whether the final value is sensible.
Like terms have exactly the same variable part and powers. Thus 3x and −7x are like terms, while 3x and 3x² are not. Constants are like terms with other constants.
To simplify, combine coefficients of like terms while leaving the variable part unchanged. This is a key distinction: 3x + 5x = 8x, but 3x × 5x = 15x². Addition combines like terms; multiplication combines factors and adds indices when the base is the same.
4x + 7 − 2x + 3 = (4x − 2x) + (7 + 3) = 2x + 10.
5a² − 3a + 2a² + 8a = 7a² + 5a.
3xy + 5x − 2xy − x = xy + 4x.
A common error is to combine unlike terms. x + x² cannot become 2x² or 2x. It is already simplified unless it is part of a factorisation or equation where another method applies.
The distributive law states that a(b + c) = ab + ac. Every term inside the bracket must be multiplied by the factor outside. With a negative factor, signs must be handled carefully.
3(2x − 5) = 6x − 15.
−2(4x + 3) = −8x − 6.
(x + 4)(x + 3) = x² + 3x + 4x + 12 = x² + 7x + 12.
(2x − 1)(x + 5) = 2x² + 10x − x − 5 = 2x² + 9x − 5.
For two brackets, multiply every term in the first bracket by every term in the second. A useful mental check is that two linear brackets should normally produce a quadratic term, a linear term and a constant.
Special identities are also useful: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b². These identities are especially important when expanding or factorising expressions efficiently.
Factorisation rewrites an expression as a product of factors. It is the reverse process of expansion and is essential for simplifying algebra, solving equations and handling quadratic expressions.
Start by looking for a common factor. For example, 6x² + 9x = 3x(2x + 3). The greatest common factor is usually the best starting point.
Factorise x² + 7x + 12.
Find two numbers whose product is 12 and whose sum is 7: 3 and 4.
Therefore x² + 7x + 12 = (x + 3)(x + 4).
For expressions of the form x² + bx + c, find two numbers with product c and sum b. For ax² + bx + c, where a is not 1, a systematic method such as splitting the middle term can be used.
Factorise 2x² + 7x + 3.
Multiply a × c = 2 × 3 = 6. Two numbers with product 6 and sum 7 are 6 and 1.
2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Also recognise the difference of two squares: a² − b² = (a − b)(a + b). For example, 9x² − 25 = (3x − 5)(3x + 5).
Index laws become increasingly important as algebra develops. For the same non-zero base, aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. Also, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. Fractional indices connect powers with roots: a¹⁄² = √a and a¹⁄³ = ∛a.
x³ × x⁵ = x⁸.
y⁹ ÷ y⁴ = y⁵.
(2a³)² = 4a⁶.
16x⁻² = 16/x², for x ≠ 0.
Surds are exact roots that cannot be simplified to rational numbers. Simplify by taking square factors outside the root: √72 = √(36 × 2) = 6√2. Surds can be added only when their simplified radical parts match, so 3√2 + 5√2 = 8√2, but √2 + √3 cannot be combined.
When rationalising a denominator such as 1/√3, multiply top and bottom by √3 to obtain √3/3. For denominators such as 2 + √3, use the conjugate 2 − √3.
Solving an equation means finding the value or values of the variable that make the statement true. The central principle is balance: whatever operation you perform on one side of an equation must be performed on the other side.
Solve 5x − 7 = 18.
Add 7 to both sides: 5x = 25.
Divide by 5: x = 5.
For equations with brackets, expand first or use an efficient inverse operation. For equations involving fractions, multiplying every term by the lowest common denominator can remove fractions cleanly.
Solve (x/3) + 2 = 7.
Subtract 2: x/3 = 5. Multiply by 3: x = 15.
Inequalities follow similar rules, but there is one critical exception: when multiplying or dividing by a negative number, reverse the inequality sign. Thus −2x > 8 gives x < −4.
When solving compound inequalities, treat all parts consistently. For example, 2 < x + 3 ≤ 7 becomes −1 < x ≤ 4.
Simultaneous equations contain two or more unknowns and are solved together. At mathematics, the main methods are elimination, substitution, and graphical solution.
Solve 2x + y = 11 and 3x − y = 9.
Add the equations: 5x = 20, so x = 4.
Substitute into 2x + y = 11: 8 + y = 11, so y = 3.
Check: 3(4) − 3 = 9 ✓
For substitution, make one variable the subject of one equation and replace it in the other. For graphical solution, the coordinates of the intersection of the two graphs give the common solution. A graph may give an approximate answer, while algebraic methods can give an exact one.
Word problems often hide simultaneous equations. Define the unknowns clearly, translate each condition into an equation, solve, then interpret the answer in context.
A quadratic equation has the general form ax² + bx + c = 0, where a ≠ 0. Unlike a linear equation, a quadratic may have two, one or no real solutions.
When possible, factorise and use the zero-product principle: if AB = 0, then A = 0 or B = 0.
Solve x² − 5x + 6 = 0.
Factorise: (x − 2)(x − 3) = 0.
Therefore x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.
For a general quadratic that does not factorise conveniently, use the quadratic formula:
Completing the square is another important quadratic technique. It rewrites a quadratic in a form that makes its structure and turning point easier to see. For example, x² + 6x + 5 = (x + 3)² − 4. This form shows that the minimum value of the expression is −4, occurring when x = −3. It can also be used to solve equations and understand the shape of quadratic graphs.
The quantity b² − 4ac is the discriminant. If it is positive, there are two distinct real roots; if it is zero, there is one repeated real root; if it is negative, there are no real roots. This provides a powerful way to predict the number of real solutions before calculating them.
For 2x² + 3x − 2 = 0, a = 2, b = 3, c = −2.
x = [−3 ± √(9 + 16)]/4 = [−3 ± 5]/4.
So x = 1/2 or x = −2.
Changing the subject means rearranging a formula so that a chosen variable is isolated on one side. The same balance principles used in equations apply.
A = πr².
Divide by π: A/π = r².
Take the square root: r = √(A/π).
For a physical length r, the positive root is normally selected.
For a formula such as v = u + at, make t the subject by subtracting u and then dividing by a: t = (v − u)/a. In more complicated formulas, clear fractions first and expand brackets only when helpful.
A sequence is an ordered list of numbers following a rule. The nth term gives the term at any position n. For an arithmetic sequence, the difference between consecutive terms is constant, so the nth term has the form an + b.
5, 8, 11, 14, … has common difference 3.
The nth term is 3n + 2 because the first term is 5 when n = 1.
The 20th term is 3(20) + 2 = 62.
Quadratic sequences have constant second differences. If the second difference is 2a, the nth-term expression begins with an². Recognising this pattern can turn a long list of calculations into a general rule.
Geometric sequences multiply by a constant ratio rather than adding a constant difference. For example, 3, 6, 12, 24, … has ratio 2. At higher mathematics difficulty, exponential expressions and geometric patterns connect sequences with indices and functions.
An algebraic equation can represent a graph. The straight-line form y = mx + c is especially important. Here m is the gradient and c is the y-intercept.
| Feature | Meaning | How to identify it |
|---|---|---|
| Gradient m | Rate of change of y with respect to x | m = (y₂ − y₁)/(x₂ − x₁) |
| y-intercept c | Value of y when x = 0 | Read c from y = mx + c |
| x-intercept | Value of x when y = 0 | Set y = 0 and solve |
| Parallel lines | Same gradient | m₁ = m₂ |
A line passes through (2, 5) and (6, 13).
Gradient m = (13 − 5)/(6 − 2) = 8/4 = 2.
Use y = 2x + c and substitute (2, 5): 5 = 4 + c, so c = 1.
The equation is y = 2x + 1.
Graphs can solve equations visually. The intersection of two graphs represents values satisfying both equations. A curve such as y = x² has symmetry and can intersect a straight line at zero, one or two points depending on the equations.
A function assigns an output to an input according to a rule. The notation f(x) means “the value of the function f when the input is x”; it does not mean f multiplied by x.
If f(x) = 3x − 4, then f(5) = 3(5) − 4 = 11.
To solve f(x) = 8, set 3x − 4 = 8, giving x = 4.
Functions can be combined. fg(x) means f(g(x)) in many mathematics contexts, so the output of g becomes the input of f. The order matters: f(g(x)) is generally not the same as g(f(x)).
An inverse function reverses the original mapping. To find it, write y = f(x), interchange x and y, then rearrange to make y the subject. The inverse exists as a function when the original mapping is one-to-one on the stated domain.
Algebraic fractions follow the same structural rules as numerical fractions. Factorise before cancelling, and cancel only factors, never separate terms joined by addition or subtraction.
(x² − 9)/(x² + 5x + 6)
Factorise: [(x − 3)(x + 3)]/[(x + 2)(x + 3)].
Cancel the common factor x + 3 to obtain (x − 3)/(x + 2), with restrictions x ≠ −3 and x ≠ −2 from the original denominator.
To add algebraic fractions, use a common denominator. For example, 1/x + 1/(x + 2) becomes [(x + 2) + x]/[x(x + 2)] = (2x + 2)/[x(x + 2)], where x ≠ 0 and x ≠ −2.
Algebra is often used to express how one quantity changes with another. In direct variation, y is proportional to x, written y ∝ x, so y = kx. In inverse variation, y ∝ 1/x, so y = k/x. Powers may also occur: y ∝ x² gives y = kx².
Suppose y ∝ x² and y = 20 when x = 2.
20 = k(2²) = 4k, so k = 5.
Therefore y = 5x². When x = 3, y = 45.
Modelling problems require you to decide what the variables represent, translate the wording into algebra, solve the mathematical model, and then interpret the result. A mathematically correct solution can still be inappropriate if it ignores context—for example, a length cannot normally be negative.
At higher difficulty, the main challenge is often not knowing a formula but deciding which algebraic structure is hidden inside the problem. Look for relationships rather than individual numbers.
When you finish an algebraic problem, ask three questions: Does the algebra balance? Does the answer satisfy the original condition? Does the answer make sense in context? This habit catches a surprising number of examination errors.
The strongest mathematics algebra students do not memorise isolated procedures. They recognise connections.
| If you see… | Think about… | Typical action |
|---|---|---|
| Several like terms | Simplification | Collect coefficients |
| Brackets | Expansion or factorisation | Distribute or take out factors |
| x² and a product/sum pattern | Quadratic structure | Factorise or use a formula |
| Two unknowns | Simultaneous equations | Eliminate or substitute |
| A comparison < or > | Inequality | Solve, reversing sign when dividing/multiplying by a negative |
| y = mx + c | Straight-line graph | Use gradient and intercept |
| f(x) | Function | Substitute, compose or invert |
| Repeated differences | Sequence pattern | Find the nth term |
| “proportional to” | Variation | Introduce a constant k |
| Fractions containing x | Algebraic fraction | Factorise and use a common denominator |
The progression is deliberate: language → manipulation → equations → functions → graphs → modelling. If the basic skills are secure, advanced questions become combinations of familiar ideas rather than completely new problems.
This article is original EDUSAMBAM educational writing. It is designed as a broad mathematics algebra resource covering the core ideas a student needs for confident algebraic manipulation, equation solving, graphical reasoning and mathematical problem-solving. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: learn the meaning of each algebraic idea, study the worked examples, practise without looking at the solution, then use examination-style questions to combine several skills in one problem.
20 questions covering basic, intermediate and advanced mathematics algebra. Answer every question, then submit to see your score instantly.