Variables, Expressions, and Equations
The core language of algebra — what variables represent, how expressions are built, and how equations are solved.
Variables
A variable is a symbol (usually a letter) that represents an unknown or changing quantity. It’s a placeholder — a name for something you don’t know yet or that can take different values.
x + 3 = 7 → x is unknown, solve for it
y = 2x + 1 → x can vary, y changes with it
The same letter can mean different things in different contexts. That’s fine. What matters is what it means within the problem you’re solving.
Expressions
An expression is a combination of numbers, variables, and operations — but with no equals sign. It has a value, but doesn’t make a claim.
3x + 5
x² − 2x + 1
4(a + b)
Terms and coefficients
An expression is made of terms separated by + or −:
3x² − 5x + 7has three terms: 3x², −5x, and 7- Coefficient: the number multiplying the variable (3 in 3x²)
- Constant term: a term with no variable (7)
- Like terms: same variable and exponent — can be combined
3x + 5x = 8x (like terms)
3x + 5x² ≠ 8x² (unlike terms — can't combine)
Simplifying expressions
Combine like terms, apply the distributive property:
2(3x + 4) − x = 6x + 8 − x = 5x + 8
Distributive property: a(b + c) = ab + ac
Equations
An equation is a statement that two expressions are equal. It makes a claim — which is either true or false, or true only for certain values of the variable.
2x + 3 = 11 → true only when x = 4
x² = 9 → true when x = 3 or x = −3
2 + 2 = 4 → always true (identity)
x + 1 = x → never true (contradiction)
Solving linear equations
The goal: isolate the variable on one side. Whatever you do to one side, do to the other.
2x + 3 = 11
2x = 8 (subtract 3 from both sides)
x = 4 (divide both sides by 2)
More complex:
3(x − 2) = 2x + 5
3x − 6 = 2x + 5 (distribute)
x − 6 = 5 (subtract 2x)
x = 11 (add 6)
Solving for a variable in a formula
Same principle — treat everything else as a constant:
v = u + at solve for t:
at = v − u
t = (v − u)/a
Systems of Equations
Two equations, two unknowns. Three methods:
Substitution — solve one equation for one variable, plug into the other:
x + y = 10
x − y = 4
From eq 1: x = 10 − y
Substitute: (10 − y) − y = 4 → 10 − 2y = 4 → y = 3
Then x = 10 − 3 = 7
Elimination — add or subtract equations to cancel a variable:
2x + 3y = 12
2x − y = 4
─────────────
4y = 8 (subtract)
y = 2, then x = 3
Graphical — each equation is a line; the solution is the intersection point. Useful for intuition, not for exact answers.
The Algebra of Manipulation
Three properties underlie all equation solving:
| Property | Statement |
|---|---|
| Reflexive | a = a |
| Symmetric | if a = b, then b = a |
| Transitive | if a = b and b = c, then a = c |
And the operations you can apply to both sides:
- Add or subtract the same quantity
- Multiply or divide by the same non-zero quantity
- Apply any function to both sides (take √ of both sides, etc.)
The trap with multiplication: multiplying both sides by an expression that could be zero can introduce false solutions. Always check solutions in the original equation.
Word Problems — Translating to Algebra
The skill is converting English into symbols:
| English | Algebra |
|---|---|
| ”is”, “equals”, “gives” | = |
| “more than”, “increased by” | + |
| “less than”, “decreased by” | − |
| “times”, “of”, “product” | × |
| “divided by”, “per”, “ratio” | ÷ |
| “what”, “how many”, “a number” | x (or any variable) |
"Three more than twice a number is 17"
2x + 3 = 17
x = 7
The discipline: define your variable clearly first. “Let x = the number of hours.” Ambiguity in the setup leads to wrong equations even with correct algebra.