What is Algebra?
Why do mathematicians use letters? Discover how variables, constants, expressions and equations work together, and why algebra is just arithmetic with mystery boxes.
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I'm thinking of a number. If I add 7 to it, the answer is 15. What's the number? Write down how you figured it out, there's more than one way.
Algebra is a way of writing maths using letters to stand for numbers. A letter used this way is a pronumeral. A pronumeral may stand for one fixed but unknown value, or for more than one possible value; when it can vary, it is also called a variable. For example, $x$ has one value in $x+5=12$, but varies through many values in the expression $x+5$.
An algebraic expression combines numbers and algebraic symbols using arithmetic operations. For example, $3x-4$ combines the pronumeral $x$ with multiplication and subtraction.
Every algebra problem has three building blocks: a variable (the unknown letter), a constant (a fixed number), and either an expression (a phrase) or an equation (a sentence with an equals sign).
Know
- What a variable and constant are
- The difference between an expression and an equation
- What a coefficient is
Understand
- Why letters are used in algebra
- That a variable can represent any number
- Why an equation needs an equals sign
Can Do
- Identify variables, constants and coefficients
- Tell expressions apart from equations
- Write simple algebraic phrases
Wrong: "$2x$ means $2 + x$"
Right: $2x$ means $2 \times x$. A number next to a letter always means multiplication.
Wrong: "$x = 5$ is an expression"
Right: $x = 5$ is an equation because it has an equals sign. Expressions never have one.
Think of a variable as a mystery box with a number inside. You can't see the number yet, but you can still do maths with the box. When you finally open it (solve the equation), you find the number.
A variable is just a container. We label it with a letter so we can talk about it. A constant is a number that sits by itself. Together they make an expression a recipe for calculating a value.
Look at the diagram above. $3x + 5$ is an expression it's just a phrase, like "three boxes plus five." But $3x + 5 = 8$ is an equation it's a complete sentence that claims two things are equal. The equals sign is what makes the difference.
The single most important distinction in beginning algebra: does it have an equals sign? If yes, it's an equation. If no, it's an expression.
An expression is like a recipe, it tells you how to calculate something. An equation is like a puzzle, it tells you two things are the same, and you need to find the missing piece.
Every algebraic expression is built from terms. Each term has parts you need to name. Being able to point at $5x$ and say "the coefficient is 5 and the variable is $x$" is a superpower.
Look at an expression like $4x + 3y - 7$. It has three terms: $4x$, $3y$, and $-7$. In $4x$, the coefficient is 4 and the variable is $x$. The term $-7$ is just a constant no variable at all.
Algebra started because people got tired of writing "some number" over and over. Here's how everyday phrases become algebra.
Look for key words. "Sum" and "more than" mean $+$. "Difference" and "less than" mean $-$. "Product" and "times" mean $\times$.
| Words | Operation | Algebra |
|---|---|---|
| A number increased by 4 | addition | $n + 4$ |
| 5 less than a number | subtraction | $n - 5$ |
| Triple a number | multiplication | $3n$ |
| A number divided by 2 | division | $n \div 2$ or $\frac{n}{2}$ |
Watch Me Solve It · Worked example
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1Count the terms$5a$, $-3b$, and $+7$ = 3 termsTerms are separated by + and - signs. Each sign belongs to the term that follows it.
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2Find the coefficient of $a$Coefficient of $a$ = 5In the term $5a$, the number multiplying $a$ is 5. That's the coefficient.
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3Identify the constant termConstant term = $+7$ (or just 7)The term $+7$ has no variable, it's just a number. That makes it the constant term.
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4Variable in the second termSecond term is $-3b$, so the variable is $b$The second term is $-3b$. The letter part is $b$, so that's the variable. Don't forget the minus sign belongs to the term!
Key Definitions
- Variable, a letter that stands for an unknown number
- Constant, a fixed number that doesn't change
- Coefficient, the number multiplying a variable
- Term, a single part of an expression, separated by + or -
Expression vs Equation
- Expression: NO equals sign (e.g. $3x + 5$)
- Equation: HAS equals sign (e.g. $3x + 5 = 20$)
- Expression = phrase, Equation = sentence
Word Clues
- "a number" → use $n$ or $x$
- "sum", "more than", "plus" → $+$
- "difference", "less than" → $-$
- "product", "times" → $\times$
Hidden Coefficients
- $x$ means $1x$ (coefficient is 1)
- $-x$ means $-1x$ (coefficient is $-1$)
- Always write the number first: $3x$, not $x3$
How are you completing this lesson?
Brain Trainer · 4 problems
Four problems to test your understanding. Work each one, then reveal the answer to check.
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1 Is $4x + 9 = 25$ an expression or an equation? Explain why.
Equation it has an equals sign, which makes it a statement that two things are equal.Equation -
2 In the expression $7m - 2n + 4$, what is the coefficient of $n$?
The term with $n$ is $-2n$. The coefficient is $-2$. Don't forget the minus sign belongs to the term!$-2$ -
3 Write an algebraic expression for: "5 more than triple a number."
Let $n$ = the number. Triple = $3n$. 5 more than = $+5$. So: $3n + 5$$3n + 5$ -
4 How many terms are in $3x + 2y - z + 8$? Name the constant term.
4 terms: $3x$, $2y$, $-z$, and $+8$. The constant term is $8$ (the term with no variable).4 terms; constant = 8
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. Explain the difference between a variable and a constant. Give one example of each.
Q7. For the expression $6a - 2b + 9$, identify: (a) the number of terms, (b) the coefficient of $a$, (c) the constant term.
Q8. Write algebraic expressions for the following: (a) Double a number increased by 3. (b) The difference between a number and 8, divided by 2.
Quick Check
1. B$3x + 7 = 22$ is an equation because it has an equals sign.
2. C The coefficient of $y$ is $-3$ (the term is $-3y$).
3. A"4 less than $n$" means $n - 4$.
4. D4 terms: $2a$, $3b$, $-c$, and $+7$.
5. B The coefficient of $-x$ is $-1$.
Show Your Working Model Answers
Q6 (2 marks): A variable is a letter that stands for an unknown or changeable number (e.g. $x$, $n$) [1 mark]. A constant is a fixed number that does not change its value (e.g. $5$, $-3$) [1 mark].
Q7 (3 marks): (a) 3 terms: $6a$, $-2b$, and $+9$ [1 mark]. (b) The coefficient of $a$ is 6 [1 mark]. (c) The constant term is 9 [1 mark].
Q8 (4 marks): Let $n$ = the number. (a) Double = $2n$, increased by 3 = $+3$, so $2n + 3$ [2 marks]. (b) Difference between $n$ and 8 = $n - 8$, divided by 2 = $\frac{n - 8}{2}$ or $(n - 8) \div 2$ [2 marks]. Accept $\frac{n-8}{2}$.
The Expression Puzzle
The expression $ax + b$ has value 11 when $x = 2$, and value 23 when $x = 5$. Find the values of $a$ and $b$.
Reveal solution
When $x = 2$: $2a + b = 11$ (1). When $x = 5$: $5a + b = 23$ (2). Subtract (1) from (2): $3a = 12$, so $a = 4$. Substitute into (1): $8 + b = 11$, so $b = 3$.
Variable
A letter that stands for an unknown number
Constant
A fixed number that never changes
Coefficient
The number multiplying a variable
Expression
A maths phrase with NO equals sign
Equation
A maths sentence WITH an equals sign
Term
A single part separated by + or -
Interactive: Algebra Machine
Substitute numbers into algebraic expressions and see them evaluate step by step.
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