Substitution into Formulae and Expressions
Substitution is a core part of algebra that helps us take general rules (formulae) and make them specific by plugging in numbers. It’s the key to solving real-world problems, from calculating a taxi fare to working out the energy of a moving object.
What’s Substitution All About?
In maths, substitution means replacing the letters (variables) in a formula or expression with specific numbers. Once you’ve substituted, you then evaluate the calculation to find a final numerical answer.
The Golden Rule: BIDMAS!
After substituting, you’ll be left with a calculation. You must follow the order of operations (BIDMAS) to evaluate it correctly and get the right answer.
Worked Examples
Example 1: Basic Substitution
If $x = 5$, evaluate $3x + 7$.
- Substitute: Replace $x$ with 5.
$3 \times (5) + 7$ - Evaluate (BIDMAS): Multiply first, then add.
$15 + 7 = 22$.
Answer: 22
Example 2: With Negative Numbers
If $y = -2$, evaluate $5y – 8$.
- Substitute: Use brackets for negatives.
$5 \times (-2) – 8$ - Evaluate (BIDMAS): Multiply first, then subtract.
$-10 – 8 = -18$.
Answer: −18
Example 3: With Powers
If $p = -3$, evaluate $p^2 – 1$.
- Substitute: Use brackets around the negative number being squared.
$(-3)^2 – 1$ - Evaluate (BIDMAS): Indices first: $(-3) \times (-3) = 9$.
$9 – 1 = 8$.
Answer: 8
Example 4: Into a Formula
Using $A = \frac{1}{2}bh$, find the area of a triangle with base $b = 12$ m and height $h = 5.5$ m.
- Substitute:
$A = \frac{1}{2} \times 12 \times 5.5$ - Evaluate:
$\frac{1}{2} \times 12 = 6$, then $6 \times 5.5 = 33$.
Answer: $33 \text{ m}^2$
Tutor Insights
🤔 Common Misunderstandings
- What $3x$ means: Forgetting that $3x$ means “3 multiplied by $x$”.
- Negative number rules: Confusing the rules for multiplying/dividing negatives, or squaring a negative (e.g., $(-5)^2 = 25$, not −25).
- BIDMAS confusion: Forgetting the order of operations after substituting.
📝 Common Exam Mistakes
- Calculation errors: Simple arithmetic mistakes, especially with negatives.
- Not showing working: You must show the substitution step and your calculations to get method marks.
- Not using brackets for negatives: Writing $-2^2$ instead of $(-2)^2$ gives −4 instead of the correct answer of 4.
Practice Questions
- Calculate the value of $7a – 4$ when $a = 3$.
- Find the value of $m^2 + 2m$ when $m = -4$.
- If $p = 6$ and $q = -1$, find the value of $3p + 5q$.
- Using $P = 2(l + w)$, find the perimeter of a rectangle with length $l = 8.2$ cm and width $w = 4.5$ cm.
Show Answers
- Working: $7(3) – 4 = 21 – 4 = \mathbf{17}$.
- Working: $(-4)^2 + 2(-4) = 16 – 8 = \mathbf{8}$.
- Working: $3(6) + 5(-1) = 18 – 5 = \mathbf{13}$.
- Working: $P = 2(8.2 + 4.5) = 2(12.7) = \mathbf{25.4 \text{ cm}}$.
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