Substitution
An expression with a letter in it is a question waiting for a number. Substitution is putting the number in. Done carefully, with brackets, it is the most reliable thing in algebra: there is a right answer and a way to check it.
National curriculum KS3: substitute numerical values into formulae and expressions, including scientific formulae; use the order of operations with brackets and powers; work with negative numbers1 · The move
Every substitution is the same three moves. Copy the expression. Replace each letter with its number, inside brackets. Calculate, following the order of operations. The brackets are not optional decoration: they are what stops 3x from becoming 3−4 when x = −4.
2 · Why the order matters
Once the number is in, you have an arithmetic problem, and arithmetic has a fixed order. UK schools call it BIDMAS (or BODMAS); in Spain it's the jerarquía de operaciones. Same rules, same reason: without an agreed order, 2 + 3 × 4 could be 20 or 14, and only one of those is right.
In substitution the step people skip is I. Faced with 3x² with x = 4, they do 3 × 4 = 12 then square it to get 144. But the power only belongs to the x: 4² = 16 first, then × 3 = 48. Watch the evaluator highlight each step in order.
3x² and (3x)² are different. In 3x² the index sits on the x alone: square the x, then multiply by 3. In (3x)² the bracket is squared, so the 3 gets squared too (that's 9x²). With x = 4: 48 versus 144. Three times smaller. The bracket is the only difference on the page, and it changes everything.
3 · Negative numbers: where marks are lost
With positive numbers, substitution is forgiving. With negatives it isn't, and every exam includes at least one. The bracket habit from Step 1 is your protection. Here are the four situations, side by side, with x = −3.
| Expression | With brackets | Working | Answer | Common wrong answer |
|---|---|---|---|---|
| x² | (−3)² | (−3) × (−3): two negatives make a positive | 9 | −9 |
| −x² | −(−3)² | square first (9), then put the minus in front | −9 | 9 |
| x³ | (−3)³ | (−3)(−3)(−3) = 9 × (−3): odd power stays negative | −27 | 27 |
| 5 − x | 5 − (−3) | subtracting a negative is adding | 8 | 2 |
| 2x | 2(−3) | positive × negative | −6 | −1 (from 2 − 3) |
| x − 4 | (−3) − 4 | going further down | −7 | 1 |
Try any expression with any negative. The evaluator shows each sign decision as it makes it.
4 · Formulae: more than one letter
A formula is an expression with a name on the left: A = lw, v = u + at. Substituting into one is the same move, repeated for each letter. Write the formula, write the values underneath, swap them all in at once with brackets, then calculate. Real formulae from science and geometry work best here, because the answer means something.
Units go in too. If u is in m/s, a in m/s² and t in seconds, the answer v comes out in m/s. In science exams the unit on the answer line is a mark. Mixing units (minutes with m/s²) gives a number that's wrong even if the arithmetic is right.
5 · Substitution in both directions: tables and graphs
Substitute several values in a row and you get a table of values, which is how every graph in KS3 is drawn. Change the expression and watch the row of answers; each column is one substitution.
6 · Check yourself
A random expression, a random value. Type your answer. The working is hidden until you commit to a number, which is how exams work too.
7 · Where substitution lives
Science
Every physics calculation is a substitution: F = ma with m = 2 kg, a = 3 m/s². Chemistry's moles = mass ÷ Mr. Biology's magnification = image ÷ actual. The formula is given; the marks are for the substitution and the unit.
Spreadsheets
A cell formula like =A2*0.2+5 is substituted into automatically for every row. Change A2 and the result updates: the spreadsheet is doing the x = … for you.
Programming
def f(x): return 3*x**2 - 2 and then f(-4) is literally substitution: the computer puts −4 where x was and follows the order of operations (Python knows BIDMAS). Try it: the answer is 46, not −50.
Graphs
To draw y = x² − 4 you substitute x = −3, −2, …, 3 and plot the seven points. Graph-drawing software does thousands of substitutions per second to make the curve look smooth.
Money
Total cost = 25 + 0.12m for a phone contract with m minutes. Substitute your monthly minutes to compare tariffs. Interest: A = P(1 + r)ⁿ with your own P, r and n.
Checking your algebra
Not sure whether 3(x + 2) really equals 3x + 6? Substitute x = 5 into both: 21 and 21. Substitute into 3x + 2 instead: 17. A quick substitution catches a wrong expansion in seconds.
Worked examples
Find the value of 5x − 2 when x = −3
Find 2a² − b when a = −2 and b = 5
Use v = u + at to find v when u = 12, a = −9.8, t = 2
Exam-style: p = 3, q = −4. Work out (p − q)² ÷ (p + q)
Quick check
What you've learned
Substitution is copy, replace with brackets, calculate. The calculation follows BIDMAS, and the step most often skipped is indices: in 3x² only the x is squared, while in (3x)² the whole bracket is. Negative values are where marks go, and the bracket habit prevents nearly every error: (−3)² is 9, −(−3)² is −9, 5 − (−3) is 8. Formulae are the same move with several letters, and a table of values is the same move repeated, which is how graphs get drawn. When in doubt about an algebraic identity, substitute a number into both sides and see.