Algebraic expressions & sequences · Ages 12–13
Substitution and input–output machines
Swap in the number and put back the ×; to go back, do the opposite.
How to do it
Method
Swap in the number and put back the ×; to go back, do the opposite
2x + 3, x = 4: 2 × 4 + 3 = 11
Another way
Backwards for the input
Output 11: 11 − 3 = 8, 8 ÷ 2 = 4
In full
To substitute, swap the letter for the number, put the × back, then work in order: powers, then times and divide, then plus and minus. With x = 3, 2x2 = 2 × 32 = 2 × 9 = 18, not 62 = 36. An input–output machine is a rule: put in a number (the input), work the rule, and out comes a number (the output). If the rule is 'times 2, then add 3', the output is 2x + 3. Each input has just one output, and such a relationship is called a function, written f(x) = 2x + 3; f(4) is the output for input 4: 2 × 4 + 3 = 11. To find the input from the output, work backwards: from the last step, do the opposite (output 11: take 3, then halve, giving 4). Or try numbers in the rule until one gives 11. Both ways give the same answer.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
When x = 11, 3x + 7 =
Show the answer and working
Answer: 40
Put the × back: 3 × 11 + 7 = 33 + 7 = 40.
Example 2
When a = 10 and b = 8, 3(a + b) − ab =
Show the answer and working
Answer: −26
ab is a × b: 3 × (10 + 8) − 10 × 8 = 3 × 18 − 80 = 54 − 80 = −26.
Example 3
All the outputs in this table come from y = 2x + 9.
True or false?
Show the answer and working
Answer: False
Put each one in: 11, 13 and 15 fit; with x = 4, y = 2 × 4 + 9 = 17, and the table shows 19.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.When x = 6, 6x + 5 =
Answer:
2.When a = 8 and b = 7, 7(a + b) − ab =
Answer:
3.The table follows one rule. What number is the '?'?
- A. 38
- B. 49
- C. 42
- D. 43
Choose the right answer
Answer:
4.When x = 6, 3x2 =
Answer:
5.This table is a function: each input has just one output.
True or false?
Answer:
All the answers
- 1. 41
- 2. 49
- 3. D. 43
- 4. 108
- 5. False
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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