Substituting into an expression

How to put a number in for a letter in an expression and calculate the value — including negative numbers and brackets.

Put a number in the box

An expression is a recipe. Substituting is cooking from it: you put a concrete number in for the letter and calculate the result. That result is called the value of the expression.

Take the expression 3x + 2 and substitute x = 5. Everywhere x stood, a five now stands: 3 · 5 + 2 = 15 + 2 = 17. The value of the expression for x = 5 is 17.

It sounds simple — and it is simple, as long as you substitute nice positive numbers. It gets interesting with negative numbers and powers. That is where most mistakes are born, so today’s main rule is: always substitute a negative number in brackets.

Substituting is the base of everything that comes: checks of equations, formulae for perimeters and areas, tables of values. Anyone who substitutes reliably has solid ground under their feet in year 8.

Jonáš is playing “guess my rule” with dad. Dad has an expression in his head and Jonáš tells him numbers. “Give me three.” — “Eleven.” — “Give me five.” — “Seventeen.” Jonáš writes: 3 → 11, 5 → 17. Difference of inputs 2, difference of outputs 6, so it is multiplied by three… and two is added to 3 · 3 = 9. “Your rule is 3x + 2!” Dad laughs: “You got it. And now the other way — I guess yours.” The whole game is powered by one skill: substitute and calculate.

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Before you start calculating, rewrite the expression with the substituted number in brackets: for x = −4 write 3 · (−4) + 2. Only then calculate. One extra line, half as many mistakes.

How to substitute step by step

1. Rewrite the expression and write the number in brackets instead of the letter. 5x − 3 for x = 2 → 5 · (2) − 3. With positive numbers the brackets can be left out, with negative ones never.

2. Fill in the hidden times. 5x is 5 · x, so after substituting 5 · 2.

3. Calculate in the order of operations. Powers first, then times and divide, finally plus and minus. 5 · 2 − 3 = 10 − 3 = 7.

With a negative number: 3x + 2 for x = −4 → 3 · (−4) + 2 = −12 + 2 = −10. Without brackets, 3 · −4 would appear on paper — and that is a notation you get lost in.

With a power: x² for x = −5 → (−5)² = 25. The brackets here decide the sign! Anyone who writes −5² = −25 substituted wrongly.

More variables: 2a + 3b for a = 4, b = −1 → 2 · 4 + 3 · (−1) = 8 − 3 = 5. Each letter has its own number, do not mix them up.

Example 1: 4x − 7 for x = 6

I rewrite with the substitution: 4 · 6 − 7.

Times first: 4 · 6 = 24.

Then minus: 24 − 7 = 17.

The value of the expression 4x − 7 for x = 6 is 17. The whole writing on paper has three lines: expression, substitution, result. Keep this rhythm — you will need it exactly the same for checking an equation.

Example 2: x² − 2x for x = −3

I rewrite with brackets: (−3)² − 2 · (−3).

Power first: (−3)² = 9.

Then multiplication: 2 · (−3) = −6. In the expression there is a minus in front of it: − (−6) = +6.

Together: 9 + 6 = 15.

Two traps in one example: (−3)² is plus nine, and “minus minus six” is plus six. Both traps were disarmed by the brackets from the rewrite.

Example 3: the perimeter of a rectangle from a formula

The formula for the perimeter of a rectangle is o = 2(a + b). The garden has a = 12 m, b = 8.5 m.

I substitute: o = 2 · (12 + 8.5).

Brackets first: 12 + 8.5 = 20.5.

Then times: 2 · 20.5 = 41 m.

A formula is just an expression with a name. Substituting into formulae is the same skill — and you will use it in geometry and in physics all year.

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The biggest trap: substituting −3 into x² without brackets. (−3)² = 9, but −3² = −9. When you substitute a negative number, the brackets are not decoration — they change the result.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Substitute the given x into the expression and calculate the value. Write only the resulting number.

Practise the questions

On paper

Three lines per question: expression, substitution in brackets, result.

Calculate the value of 2x + 9 for x = 3, for x = 0 and for x = −4. Write each substitution on its own line.
Calculate the value of x² + x for x = 5 and for x = −5. Write in a sentence why the results are not the same.
Into the formula s = v · t substitute v = 15 km/h and t = 2.5 h. How many kilometres does the cyclist ride?
Invent an expression with one variable, let a classmate or a parent say a number and calculate the value. Then swap roles — like Jonáš with dad.

Now you 💪

I always substitute negative numbers in brackets.
I keep the order of operations: power, then times, then plus and minus.
I write the substitution as its own line, not from my head.

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Done when: You have calculated six values including negative substitutions, one physics formula, and a run of five correct in practice.

What to take from this lesson

Substituting = put a number in for the letter, calculate the value of the expression.
A negative number belongs in brackets when you substitute. Always.
Order of operations: powers → times and divide → plus and minus.
A formula is an expression with a name; you substitute into it in exactly the same way.
Writing in three lines (expression, substitution, result) protects you from mistakes.