Equations and Identities
Ever wondered why some maths problems ask you to ‘solve’ for $x$, while others ask you to ‘show that’ two things are the same? It all comes down to the important difference between an equation and an identity.
Equation vs. Identity: What’s the Difference?
Equation (=)
An equation is like a balanced scale. It’s a statement that two expressions are equal, but it’s only true for specific values of the variable.
This is only true when $x = 7$.
When you see an equation, your job is to solve it to find the unknown value.
Identity (≡)
An identity is like having two different blueprints for the exact same object. It’s a statement that two expressions are equivalent — true for all values of the variable.
This is true for any value of $x$.
When you see an identity, your job is to prove it by showing one side can be transformed into the other.
Worked Examples
Solving an Equation
Solve $5x – 7 = 3x + 1$.
Here, we find the specific value of $x$ that makes the statement true.
- Collect $x$ terms:
$5x – 3x – 7 = 1 \implies 2x – 7 = 1$ - Collect number terms:
$2x = 1 + 7 \implies 2x = 8$ - Isolate $x$:
$x = 8 \div 2$
Answer: $x = 4$
Proving an Identity
Show that $(x+3)(x-2) \equiv x^2 + x – 6$.
Here, we transform the left-hand side (LHS) until it matches the right-hand side (RHS).
- Start with the LHS:
$(x+3)(x-2)$ - Expand (FOIL):
$x^2 – 2x + 3x – 6$ - Collect like terms:
$x^2 + x – 6$ ≡ RHS ∎
Proving a More Complex Identity
Prove that $(n+5)^2 – (n-5)^2 \equiv 20n$.
- Expand each bracket:
$(n^2 + 10n + 25) – (n^2 – 10n + 25)$ - Distribute the negative sign carefully:
$n^2 + 10n + 25 – n^2 + 10n – 25$ - Collect like terms:
$(n^2 – n^2) + (10n + 10n) + (25 – 25) = 20n$ ≡ RHS ∎
Tutor Insights
🤔 Common Misunderstandings
- Trying to “solve” an identity. An identity doesn’t have a single solution for $x$ — it’s a rule you prove.
- Using ‘=’ instead of ‘≡’ in proofs. Using the identity symbol shows you understand the concept.
- Working on both sides at once. To prove an identity, you must transform one side until it matches the other.
📝 Common Exam Mistakes
- Errors in algebraic manipulation, especially expanding brackets or handling negative signs.
- Not showing enough working. You must show the clear, logical steps that get you from one side to the other.
- Not fully simplifying and stopping before the two sides match.
Practice Questions
- (Equation) Solve: $4y + 11 = 35 – 2y$.
- (Identity) Prove: $5(x + 2) – 3(x + 1) \equiv 2x + 7$.
- (Equation) Solve: $x^2 + 7x + 10 = 0$.
- (Identity) Prove that $(k + 1)^2 – (k – 1)^2 \equiv 4k$.
Show Answers
- Working: $4y + 2y = 35 – 11 \implies 6y = 24$.
Answer: $y = 4$. - Proof: LHS $= 5x + 10 – 3x – 3 = 2x + 7$ ≡ RHS ∎
- Working: Factorise: $(x+2)(x+5) = 0$.
Answer: $x = -2$ or $x = -5$. - Proof: LHS $= (k^2 + 2k + 1) – (k^2 – 2k + 1) = k^2 + 2k + 1 – k^2 + 2k – 1 = 4k$ ≡ RHS ∎
FAQs
Q: Can I prove an identity by substituting values?
A: No. Substituting values can check if an identity might be true for those specific numbers, but it doesn’t prove it’s true for all possible values. A formal proof requires algebraic manipulation.
Q: Do I always have to start with the Left Hand Side (LHS)?
A: No, you can start with whichever side looks more complex and simplify it until it matches the other, simpler side. The goal is to show one side becomes the other.
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