Substituting into Formulas

What is substitution?

A formula is a rule written with letters, such as A = lw for the area of a rectangle. Substitution means swapping each letter for a number you are given, then working out the answer.

This is also called evaluating an expression. It is the reverse of writing algebraic expressions: there you turned words into letters, and here you turn letters back into numbers.

One formula can produce endless answers. The area formula gives 24 for a 6-by-4 rectangle and 50 for a 10-by-5 one — same rule, different inputs.

The golden rule: use brackets

Whenever you substitute a value, wrap it in brackets. This keeps the number together, makes multiplication clear, and protects you from sign mistakes.

For example, to evaluate 3x when x = 5, write 3(5), which clearly means 3 × 5 = 15. The brackets become essential with negatives and powers, as you'll see.

After substituting, use BODMAS

Once the letters are gone, you have an ordinary arithmetic calculation. Follow the order of operations:

Step	Meaning
B	Brackets
O	Orders (powers and roots)
DM	Division and Multiplication (left to right)
AS	Addition and Subtraction (left to right)

If you need a full refresher, see Order of Operations (BODMAS).

Worked example 1 — one variable

Find the value of 4x − 3 when x = 6.

1. Replace x with (6): 4(6) − 3.
2. Multiply first (BODMAS): 4 × 6 = 24.
3. Subtract: 24 − 3 = 21.

Worked example 2 — two variables

Find the value of 2a + 3b when a = 5 and b = 4.

1. Substitute both: 2(5) + 3(4).
2. Multiply each term: 2 × 5 = 10 and 3 × 4 = 12.
3. Add: 10 + 12 = 22.

Substitute every letter before you start calculating, and keep each substituted value in its own bracket.

Worked example 3 — a power

Find the value of x² + 2x when x = 3.

1. Substitute with brackets: (3)² + 2(3).
2. Orders first: (3)² = 9.
3. Then multiply: 2 × 3 = 6.
4. Add: 9 + 6 = 15.

The brackets around the 3 remind you to square the whole value. Doing the power before the multiplication is what BODMAS requires.

Worked example 4 — a negative value

Find the value of 5 − 2n when n = −4.

This is exactly where brackets save you.

1. Substitute: 5 − 2(−4).
2. Multiply: 2 × (−4) = −8, so the expression is 5 − (−8).
3. Subtracting a negative adds: 5 + 8 = 13.

Without brackets, "5 − 2 × −4" is easy to misread. With them, the double-negative is obvious.

Worked example 5 — a negative squared

Find the value of x² when x = −3.

1. Substitute with brackets: (−3)².
2. (−3)² means (−3) × (−3).
3. Negative times negative is positive: 9.

Compare this with −3², which by BODMAS means −(3²) = −9. The brackets are the difference between +9 and −9, so they truly matter.

Worked example 6 — a real formula

The perimeter of a rectangle is P = 2(l + w). Find P when l = 8 and w = 5.

1. Substitute: P = 2(8 + 5).
2. Brackets first: 8 + 5 = 13.
3. Multiply: 2 × 13 = 26.

So the perimeter is 26 units. Notice how the bracket in the formula forces you to add before multiplying — exactly what BODMAS would tell you anyway.

Activity — evaluate each expression

1. 3x + 7 when x = 4
2. 5a − 2b when a = 6 and b = 5
3. x² − 1 when x = 5
4. 10 − 3n when n = −2
5. Using A = lw, find A when l = 9 and w = 7
6. Using P = 2(l + w), find P when l = 10 and w = 3

Answers: 1) 19 2) 20 3) 24 4) 16 5) 63 6) 26

Why this matters

Substitution is how formulas become useful. Scientists, builders, and accountants all use formulas, and every one of them works by replacing letters with measured numbers and calculating the result. The same skill lets you check the answer to an equation by substituting it back in.

Hold on to two habits: bracket every value you substitute, and then follow BODMAS. Together they keep your signs right and your powers in the correct order, which is where almost every substitution mistake comes from.