Algebra Basics
This branch of mathematics, which uses letters or symbols to represent numbers, is very powerful as it allows statements to be made that are true for all cases, not just specific ones.
Vocabulary
Constant
A constant is a value, usually a number, which does not change.
3, 8, π
Variable
A variable is a symbol, usually a letter, standing for any value from a given range of numbers.
x, y, a, z
Coefficient
This is a constant in front of a variable. A coefficient of 1 is usually not written.
3x = 3 × x
27y² = 27 × y × y
z = 1 × z
Expression
An expression is a list of constants and variables joined by + − × ÷.
45a − 27b
15x + 9y
(15/c) × 3a
Terms
Each part of an expression is called a term.
45x + 32x² → two terms
45x + 32x² − 9 → three terms
Collect Like Terms
Like terms have the same variable.
When we collect like terms, we gather together all the terms that are the same.
x + 5 + 8 + Q + 9 + M + x + Q + x + Q + Q + Q + x
→ x + x + x + x + Q + Q + Q + Q + Q + M + 5 + 8 + 9
= 4x + 5Q + M + 22
Simplify
Collect like terms using addition or subtraction.
Order: coefficients → variables → numbers.
4x + 17x = 21x
a + 6b + 23a − 3 = 24a + 6b − 3
3a + 6b + 6 − 2a + 3b + 4 = a + 9b + 10
Evaluate
Substitute values and calculate.
Remember the order of operations.
If a = 2, b = 3:
a + 2b = 2 + 6 = 8
a² − 3b + 5 = 4 − 9 + 5 = 0
Equation
An equation sets two expressions equal.
2b + 3 = 7
3(a + b) = 3a + 3b
y = x + 2
y = mx + c
Solving EquationsInequation
Satisfy an Equation
A value satisfies an equation if it makes both sides equal.
b = 2 satisfies 2b + 3 = 7 because 2×2 + 3 = 7
b = 3 does not satisfy it because 2×3 + 3 = 9
Solution
The solution is the value that satisfies the equation.
2b + 3 = 7 → b = 2
Formulae
A formula is an equation used to calculate values.
Formula in words:
Algebra Word Problem
Problem:
A number is increased by 7 and the result is 31.
Form an equation and solve the problem using words.
Step 1: Describe the situation in words
Start with an unknown number. Add 7 to this number. The total becomes 31.
Step 2: Turn the words into an equation
“A number increased by 7 is 31” becomes \[ x + 7 = 31 \]
Step 3: Explain the solving steps in words
To find the number, undo the +7 by subtracting 7 from both sides. This leaves the unknown number on its own.
\[ x = 31 - 7 \] \[ x = 24 \]
Final Answer
The number is 24.
Tuition costs £20/hour + £5 travel.Find the cost of 6 hours worth of tuition.
6 hours → 6×20 + 5 = £125
Formula in symbols:
Reading, Writing and Solving Formulae in Symbols
In algebra, a formula is a rule written using letters and symbols. Each letter stands for a quantity, and the formula shows how those quantities are linked.
- Reading a formula means understanding what each symbol represents and how the quantities are connected. \y = mx + c ) means “to find the value of y, multiple the value of x by m and add c”.
- Writing a formula means turning a word description into algebra. “Speed equals distance divided by time” becomes \( s= \dfrac{d}{t} \).
- Solving a formula means rearranging it to find a missing value. For example, from \( s = \dfrac{d}{t} \) you can make distance the subject: \( d = vt \).
Being confident with formulae helps you move between words and symbols, choose the correct operations, and solve real‑world problems efficiently.
The cost of tuition is given by the formula C = 20h + 5 , where C= cost of tutoring (£) and h = hours of tuition.
Find the cost of 6 hours worth of tuition.
C = 20h + 5
C = 20 x 6 + 5
C = 120 +5 = £125
Evaluating Formulae
Eₖ = ½mv²
m = 5 kg, v = 12 m/s
Eₖ = ½ × 5 × 12 × 12 = 360 J
Books
Printed resources available at Amazon
Algebra Word Questions Algebra Word Questions
Algebra Word Questions (Drill Questions)
This book contains 250 algebra word questions, ranging from simple statements like “Six times a number is 78” to more complex problems involving consecutive integers and multi‑step relationships. The aim is always the same: find the number — or numbers — that make each statement true.
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