FOUNDATIONS 03
Equations and Identities
An equation asks which values make two expressions equal. An identity states that two expressions are equal for every allowed value.
The main idea
An equals sign expresses a relationship between two quantities. If a £3 starting charge plus £2 per mile gives a total cost of £13, then
\[3+2x=13.\]The equation lets us recover the unknown distance \(x\). Algebra works because every step must preserve the equality.
1. Equality means balance
The symbol \(=\) means that the expressions on its two sides have exactly the same value.
\[3+4=7.\]It does not mean “the answer comes next”. The statement \(7=3+4\) is equally valid.
Both sides represent the same value.
2. An equation may be true only for particular values
Consider
\[x+3=8.\]This is not true for every \(x\). It becomes true when \(x=5\). A value that makes an equation true is called a solution.
\(5+3=8\)
\(8=8\quad\checkmark\)
3. Solving preserves equality
To solve an equation, perform operations that keep both sides equal. For
\[2x+3=11,\]subtract \(3\) from both sides, then divide both sides by \(2\).
\(2x+3-3=11-3\)
\(2x=8\)
\(x=4\)
4. Why doing the same thing to both sides works
If \(a=b\), adding the same quantity to both sides preserves equality:
\[a+c=b+c.\]Multiplying both sides by the same quantity also preserves equality:
\[ac=bc.\]Division is valid provided we do not divide by zero.
\(\Downarrow\)
\(a+c=b+c\)
\(\Downarrow\)
equality preserved
5. Equivalent equations
Different-looking equations are equivalent when they have exactly the same solutions.
\[x+2=6\quad\Longleftrightarrow\quad x=4.\]Solving is a sequence of valid transformations into simpler equivalent equations.
\(\Downarrow\; -2\text{ on both sides}\)
\(x=4\)
6. Identities
An identity is true for every allowed value of its variable.
\[2(x+3)\equiv2x+6.\]For example, substituting \(x=1\), \(x=5\), or any other real value makes both sides equal.
\(x=5:\;2(8)=10+6=16\)
same relationship for every \(x\)
7. Equation versus identity
The distinction is important. An equation usually asks which values work; an identity describes an equality that always holds.
true only when \(x=3\)
true for every real \(x\)
8. Checking a solution
After solving an equation, substitute the result into the original equation. This verifies that no algebraic mistake has been made.
For \(2x+3=11\), using \(x=4\) gives
\[2(4)+3=11.\]\(=8+3\)
\(=11\quad\checkmark\)
An equation is a statement of equality that may hold only for particular values. Solving means transforming it without changing its solutions. An identity is stronger: it is true for every allowed value.
Why this matters for calculus
Calculus constantly uses equations and identities. We solve equations to locate intersections, stationary points and boundary values, while algebraic and trigonometric identities help rewrite expressions into forms that are easier to differentiate or integrate.