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Substitution

Maths · Y7 Algebra
Year 7
Lesson Guide
Explanation
Substitution simply means swapping a letter for a number you have been given. Picture the expression 3n plus 2 as a machine: you feed a number in and a value comes out. Feed in 1 and you get 3 times 1 plus 2, which is 5. Feed in 2 and you get 8. Feed in 4 and you get 14. Feed in 10 and you get 32. The rule inside the machine never changes; only the number going in does. There is one habit that saves marks every single time. When you replace the letter, put the multiplication sign back. So 3n with n equal to 4 becomes 3 times 4, which is 12, not the two-digit number 34.
Your Turn
Find the value of 5m + 1 when m = 6, and again when m = 20. Check: 5 × 6 + 1 = 31, and 5 × 20 + 1 = 101.
Watch Out
Writing 3n as 34 when n = 4 is the classic slip. Rewrite the expression with a × sign before you calculate anything.
Explanation
Substituting does not switch off the order of operations — it just hands you numbers to apply them to. Compare 3x plus 4 and 3 bracket x plus 4, both at x equal to 5. In the first you rewrite it as 3 times 5 plus 4. Multiplication comes before addition, so that is 15 plus 4, giving 19. In the second, brackets come first, so 5 plus 4 is 9, and 3 times 9 is 27. Same letter, same number, very different answers. Here is one more to watch: 20 minus 2x at x equal to 6 is 20 minus 12, which is 8. If you subtracted first you would get a nonsense answer.
Your Turn
Find the value of 4(y + 3) when y = 2, and of 4y + 3 when y = 2. Check: 4 × 5 = 20, and 8 + 3 = 11.
Watch Out
Work left to right only AFTER brackets and multiplying are done. In 20 − 2x, the 2x must be worked out first.
Explanation
Expressions often carry more than one letter, and each letter gets its own value. Suppose a is 7 and b is 3. Then a plus b is 10 and a minus b is 4, which is easy enough. But watch ab: the shorthand is hiding a multiplication sign, so ab means 7 times 3, which is 21, and never the number 73. For 2a plus 3b, deal with each term separately. Two a is 2 times 7, which is 14. Three b is 3 times 3, which is 9. Add them and you get 23. Take it slowly, one term at a time, and rewrite every hidden multiplication sign before you start calculating.
Your Turn
If p = 5 and q = 8, find p + q, pq and 4p − q. Check: 13, 40, and 20 − 8 = 12.
Watch Out
ab is a × b, not the two digits written next to each other. With a = 7 and b = 3 the answer is 21, not 73.
Explanation
Squares need real care, because the little 2 only belongs to the letter it is touching. Find 3x squared when x is 4. The expression says square the x, then multiply that by 3. So do the squaring first: 4 times 4 is 16. Then multiply by 3 to get 48. The tempting wrong route is to do 3 times 4 first, getting 12, and then square that to get 144. That squares the 3 as well, which the expression never asked for. The same rule holds every time. At x equal to 5, the square is 25 and three of those is 75. Powers before multiplication, always.
Your Turn
Find the value of 2x2 when x = 3, and of x2 + 1 when x = 6. Check: 2 × 9 = 18, and 36 + 1 = 37.
Watch Out
In 3x2 only the x is squared. At x = 4 the answer is 48, not 144. Do the power before the multiplication.
Explanation
This is where substitution actually earns its keep. A formula is an expression with a name and a job. The perimeter of a rectangle is P equals 2l plus 2w, so for a rectangle 8 centimetres long and 5 wide, that is 2 times 8 plus 2 times 5, which is 16 plus 10, giving 26 centimetres. The area is A equals lw, which is 8 times 5, so 40 square centimetres. Everyday life is full of these. Hiring a bike might cost 15 dollars plus 8 dollars an hour, giving C equals 8h plus 15. Four hours costs 32 plus 15, which is 47 dollars. Substitute, follow the order of operations, and remember the units.
Your Turn
Using C = 8h + 15, find the cost of hiring the bike for 3 hours. Check: 8 × 3 = 24, and 24 + 15 = 39 dollars.
Watch Out
Do not forget the fixed part of a formula. In C = 8h + 15 the 15 is charged once, no matter how many hours you hire for.
📌 Key Points — this lesson in a nutshell
  • Substitution means replacing each letter with the number you have been given.
  • Put the hidden multiplication sign back first: 3n with n = 4 becomes 3 × 4 = 12, not 34.
  • The order of operations still applies: brackets, then × and ÷, then + and −.
  • 3x + 4 and 3(x + 4) give different answers — at x = 5 they are 19 and 27.
  • Each letter gets its own value, so with a = 7 and b = 3, ab means 7 × 3 = 21.
  • In 3x2 only the x is squared. At x = 4 that is 3 × 16 = 48, not 144.
  • Do powers before multiplying, and multiplying before adding.
  • A formula such as P = 2l + 2w is just an expression you substitute into.
  • Write the units on a formula answer, such as cm, square cm or dollars.
  • Substituting a friendly number is also the best way to check an expression you wrote yourself.
Step 1: Introduction
Step 1 of 5
The substitution machine the rule inside 3n + 2 put n in value out n = 1 3 × 1 + 2 5 n = 2 3 × 2 + 2 8 n = 4 3 × 4 + 2 14 n = 10 3 × 10 + 2 32 SUBSTITUTION means replacing the letter with a given number. One rule, four different answers — the rule never changes, only the number you feed it. Put the times sign BACK when you substitute. 3n with n = 4 becomes 3 × 4 = 12. It does not become the number 34.
Both start at x = 5, but the order is different 3x + 4 rewrite it: 3 × 5 + 4 multiply first: 15 + 4 = 19 only the x was tripled 3(x + 4) rewrite it: 3 × (5 + 4) brackets first: 3 × 9 = 27 the whole bracket was tripled Order of operations still rules: brackets, then × and ÷, then + and −. Substituting does not switch those rules off — it just gives you numbers to use them on. One more: 20 − 2x when x = 6 Multiply before subtracting: 20 − 12 = 8. Doing 20 − 2 first would wrongly give 108.
When a = 7 and b = 3 a = 7 b = 3 a + b 7 + 3 = 10 a − b 7 − 3 = 4 ab 7 × 3 = 21 2a + 3b 14 + 9 = 23 5a − 2b 35 − 6 = 29 a + b + 10 7 + 3 + 10 = 20 Remember what the shorthand is hiding. ab means a × b, so it is 7 × 3 = 21 — never the number 73. 2a means 2 × a = 14, and 3b means 3 × b = 9, so 2a + 3b = 23.
Find 3x2 when x = 4 Step 1 — read it carefully: only the x is squared, not the 3. Step 2 — square first: 4 × 4 = 16 Step 3 — then multiply by 3: 3 × 16 = 48 The wrong route 3 × 4 = 12, then 12 × 12 = 144 That squares the 3 as well, which the expression never asked for. More values, same rule x = 2: square gives 4, so 3x2 = 12 x = 5: square gives 25, so 3x2 = 75 x = 10: square gives 100, so 3x2 = 300
Formulas are expressions with a job length l = 8 cm a rectangle w = 5 Perimeter: P = 2l + 2w P = 2 × 8 + 2 × 5 = 16 + 10 = 26 cm Area: A = lw A = 8 × 5 = 40 square cm Hiring a bike costs 15 dollars, plus 8 dollars for each hour: C = 8h + 15 h = 1 → 8 + 15 $23 h = 2 → 16 + 15 $31 h = 4 → 32 + 15 $47 h = 6 → 48 + 15 $63 A formula is just an expression with a name and a purpose. Substitute the numbers you are given, follow the order of operations, and add the units.
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