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AQA GCSE · Question 07.3 · Algebra
Here is a different number machine.
When x = 5, y = 13
and
when x = 10, y = 28
Complete the number machine.
Here is a different number machine.
When x = 5, y = 13
and
when x = 10, y = 28
Complete the number machine.
How to approach this question
1. Let the first operation be "× a" and the second be "- b". The rule is y = ax - b.
2. Use the two given pairs of values to form two simultaneous equations.
- For x=5, y=13: 13 = 5a - b
- For x=10, y=28: 28 = 10a - b
3. Solve these equations. A quick way is to find the difference between the inputs and outputs.
- The input x increases by 5 (from 5 to 10).
- The output y increases by 15 (from 13 to 28).
4. The multiplier (the number in the first box) is the change in output divided by the change in input.
- Multiplier (a) = 15 / 5 = 3.
5. Now we know the first box is "× 3". Use one of the pairs to find the second operation.
- Using x=5: Input is 5. After "× 3", the number is 5 × 3 = 15.
- The final output is 13. What do you do to 15 to get 13? You subtract 2.
- So the second box is "- 2".
6. Check with the other pair (x=10, y=28).
- Input 10. 10 × 3 = 30. 30 - 2 = 28. It works.
Full Answer
First box: × 3
Second box: - 2
This is a two-step function machine. Let's call the multiplication value 'm' and the subtraction value 's'. The rule is y = (x * m) - s.
We are given two pairs of (x, y) values: (5, 13) and (10, 28).
**Step 1: Find the multiplier (m)**
We can find the multiplier by looking at the change in y for a given change in x.
- Change in x = 10 - 5 = 5
- Change in y = 28 - 13 = 15
The multiplier `m` is the change in y divided by the change in x.
m = 15 / 5 = 3.
So, the first operation is **× 3**.
**Step 2: Find the subtraction value (s)**
Now we know the rule is y = 3x - s. We can use one of the pairs of values to find `s`. Let's use (x=5, y=13).
1. Input x = 5.
2. First operation: 5 × 3 = 15.
3. Second operation: 15 - s = 13 (the output).
4. To find `s`, we can rearrange: s = 15 - 13 = 2.
So, the second operation is **- 2**.
**Step 3: Check with the other pair of values**
Let's check if the rule y = 3x - 2 works for (x=10, y=28).
- Input x = 10.
- 10 × 3 = 30.
- 30 - 2 = 28.
The output is 28, which is correct.
So the completed machine is: **[× 3]** and **[- 2]**.
Common mistakes
✗ Guessing and checking without a systematic method.
✗ Getting the order of operations wrong.
✗ Finding the multiplier correctly but then making an error finding the second number.
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