1. Year 8: Changing the Subject
Dr J Frost
Last modified: 24th May 2016

2. 2𝑥+7 3 =9 𝑥=10 3 𝑥 2 +5=152 𝑥=7 𝑥= 5 2 5𝑥−2=3𝑥+3 Starter ? ? ?
Solve the following equations:
𝑥= 5 2 ?
5𝑥−2=3𝑥+3 S1
2𝑥+7 3 =9 ?
𝑥=10 S2 S3
3 𝑥 2 +5=152 ?
𝑥=7

3. 2𝑥−5 =3 2𝑥−5=9 2𝑥=14 𝑥=7 Solving Equations
Imagine the 𝑥 stuck inside a prison – we gradually have to ‘undo’ the things around it before it can be released.
Undo the last thing done to 𝒙 on each step by doing the opposite.
2𝑥−5 =3
Click
We can’t add 5 yet because it’s ‘trapped’ inside √.
We ‘square’ to undo the √.
2𝑥−5=9
Click
2𝑥=14
Click
𝑥=7

4. Test Your Understanding1
Solve 2 2𝑥+5 =3
𝟒𝒙+𝟏𝟎=𝟑 𝟒𝒙=−𝟕 𝒙=− 𝟕 𝟒 2
Solve 2 𝑥 =11 𝟐 𝒙 𝟐 +𝟏=𝟑𝟑 𝟐 𝒙 𝟐 =𝟑𝟐 𝒙 𝟐 =𝟏𝟔 𝒙=±𝟒 ? ?
Solve 3 2−2𝑦 =4 1−5𝑦
𝟔−𝟔𝒙=𝟒−𝟐𝟎𝒙 𝟏𝟒𝒙=−𝟐 𝒙=− 𝟐 𝟏𝟒 =− 𝟏 𝟕 3 ?

5. 𝐹= 5 9 𝐶−32 Changing the Subject of a Formula
The formula to calculate a temperature in Fahrenheit if we have the temperature in Celsius:
𝐹= 5 9 𝐶−32
The subject of the formula is the variable the appears on its own on one side of the equation (usually the left) and not on the other side.
But what if we had say the temperature in Fahrenheit, and wanted to know it in Celsius?

6. Skill #1: ‘Undoing’ to Unlock
Make 𝑥 the subject of the formula.
Undo the last thing done to the subject each time by doing the opposite.
Bro Tip: It doesn’t matter what side the subject is on, provided it’s on its own! ?
𝑦=𝑥−2
𝑥=𝑦+2
𝑥= 𝑦−2 3 ?
𝑦=3𝑥+2 ?
𝑦= 𝑥 +1
𝑥= 𝑦−1 2
𝑦= 𝑥 2 −𝑎 4
𝑥=± 4𝑦+𝑎 ?

7. Exercise 1
In each case make 𝑥 the subject of the formula. (Please copy out question first) ? 1
𝑥−2𝑎=𝑏 → 𝒙=𝒃+𝟐𝒂 2𝑥−𝑦=𝑧 → 𝒙= 𝒛+𝒚 𝟐 𝑥 𝑎 +𝑏=𝑐 →𝒙=𝒂 𝒄−𝒃 𝑥 −2𝑏=𝑐 → 𝒙= 𝒄+𝟐𝒃 𝟐
𝑎 𝑥+𝑏 =𝑐 →𝒙= 𝒄 𝒂 𝟐 −𝒃 2= 2𝑥−𝑎 𝑏 →𝒙= 𝟐𝒃+𝒂 𝟐
𝑧 2 =3 𝑥 →𝒙= 𝒛 𝟐 𝟑 −𝟒 4 𝑥 2 + 𝑏 2 =𝑐 →𝒙=± 𝒄− 𝒃 𝟐 𝟒
3 𝑥 − 𝑦 2 =3 →𝒙= 𝟑+ 𝒚 𝟐 𝟑 𝟐 1+ 𝑑+3𝑥 4 = 𝑒 2 →𝒙= 𝟒 𝒆 𝟐 −𝟏 −𝒅 𝟑 ?
𝑎= 1 2 𝑏+𝑥 →𝒙=𝟐𝒂−𝒃
𝑥 𝑎 = 𝑏 𝑐 →𝒙= 𝒂𝒃 𝒄
𝑎𝑥 3 + 𝑏 2 = 𝑐 →𝒙= 𝟑 𝒄 𝟑 − 𝒃 𝟐 𝒂
𝑦=𝑎+ 𝑏+ 𝑐+𝑥 →𝒙= 𝒚−𝒂 𝟐 −𝒃 𝟐 −𝒄 𝑦= 𝑥 2 −1 𝑎 3−𝑦 →𝒙=± 𝒂𝒚 𝟑−𝒚 𝟐 +𝟏
𝑎 𝑥 2 −𝑏 𝑎+𝑏 = 𝑐 2 +𝑑 →𝒙=± 𝒄 𝟐 +𝒅 𝟐 𝒂+𝒃 +𝒃 𝒂 ? 11 2 ? 3 ? 12 4 ? ? ? 13 5 ? 6 14 ? ? 7 ? N 8 ? ? 9 N ? 10 ?

8. Vote with your diaries RED ORANGE GREEN BLUE
Test your understanding so far…
RED
ORANGE
GREEN
BLUE

9. 𝑦=2𝑥−3
Make 𝒙 the subject.
𝑥=𝑦+ 3 2
𝑥= 𝑦+3 2
𝑥=2𝑦−3
𝑥= 𝑦−3 2

10. 𝑦=𝑎 𝑥
Make 𝒙 the subject.
𝑥= 𝑦 2 𝑎
𝑥= 𝑦 𝑎 2
𝑥= 𝑦 𝑎
𝑥= 𝑎𝑦 2

11. 𝑦=𝑎 𝑥 2 +1 Make 𝒙 the subject. 𝒙=± 𝒚 𝒂 −𝟏 𝒙=± 𝒚− 𝟏 𝒂 𝒙=± 𝒚−𝟏 𝒂
𝒙=± 𝒚− 𝟏 𝒂
𝒙=± 𝒚 𝒂 −𝟏
𝒙=± 𝒚−𝟏 𝒂
𝒙=± 𝒚 𝒂 −𝟏

12. 𝑦=𝑏 𝑥+1 Make 𝒙 the subject. 𝒙= 𝒚 𝒃 𝟐 −𝟏 𝒙= 𝒚−𝟏 𝟐 𝒃 𝒙= 𝒚 𝟐 𝒃 −𝟏
𝑦=𝑏 𝑥+1
Make 𝒙 the subject.
𝒙= 𝒚 𝒃 𝟐 −𝟏
𝒙= 𝒚−𝟏 𝟐 𝒃
𝒙= 𝒚 𝟐 𝒃 −𝟏
𝒙= 𝒚 𝟐 −𝟏 𝒃

13. Skill #2: Subject trapped in a negative term
When the subject is within the first argument of a subtraction, it’s easy to ‘release’.
𝑦=2𝒙−3
2𝒙=𝑦+3 ?
However, it’s a tiny bit harder if the subject is in the term being subtracted.
When the subject is inside a negative term, just add it to both sides.
𝑦=3−2𝑥
𝑦+2𝑥=3 ?
2𝑥=3−𝑦 ?

14. Examples ? ? ? 𝑎−𝑥=𝑏 𝒂=𝒃+𝒙 𝒙=𝒂−𝒃 1−𝑏𝑥=𝑐 𝟏=𝒄+𝒃𝒙 𝟏−𝒄=𝒃𝒙 𝒙= 𝟏−𝒄 𝒃2 ? ?
𝑎𝑏−𝑐 𝑥 =𝑦+1 𝒂𝒃=𝒚+𝟏+𝒄 𝒙 𝒂𝒃−𝒚−𝟏=𝒄 𝒙 𝒂𝒃−𝒚−𝟏 𝒄 = 𝒙 𝒙= 𝒂𝒃−𝒚−𝟏 𝒄 𝟐 3 ?

15. Doing it in one step… (if you like)
How could you rearrange the numbers in 9−5=4 to get another subtraction?
9−4=5 ?
This suggests you can swap the thing you’re subtracting with the result. (i.e. Only the thing to the left of the subtraction stays put)
Examples:
𝑎−𝑥=𝑏 𝒂−𝒃=𝒙
𝑏𝑐= 𝑏 2 −2𝑥 𝟐𝒙= 𝒃 𝟐 −𝒃𝒄 𝒙= 𝒃 𝟐 −𝒃𝒄 𝟐
𝑦 𝑧 2 −𝑎 𝑥 2 =𝑏𝑦 𝒂 𝒙 𝟐 =𝒚 𝒛 𝟐 −𝒃𝒚 𝒙 𝟐 = 𝒚 𝒛 𝟐 −𝒃𝒚 𝒂 𝒙=± 𝒚 𝒛 𝟐 −𝒃𝒚 𝒂 ? ? ?

16. Exercise 2
In each case make 𝑥 the subject of the formula. (Please copy out question first)
2−3𝑥 𝑟 =2𝑠 →𝒙= 𝟐−𝟐𝒓𝒔 𝟑
1− 𝑎 2 𝑏− 𝑥 2 =𝑐 →𝒙=± 𝒃− 𝟐−𝟐𝒄 𝒂
𝑦=1− 1−𝑥 2 → 𝒙=𝟏± 𝟏−𝒚
𝑎𝑏= 𝑐 2 −𝑑 𝑥 →𝒙= 𝒄 𝟐 −𝒂𝒃 𝒅 𝟐 𝑦=1− 1− 1−𝑥 →𝒙=𝟏− 𝟏− 𝟏−𝒚 𝟐 𝟐
𝑎− 𝑏−𝑐 𝑥 𝑑+𝑒 =𝑓−𝑔 →𝒙= 𝒃− 𝒂+𝒈−𝒇 𝒅+𝒆 𝒄 𝟐 ? 1
𝑏 2 −𝑥=𝑐 → 𝒙= 𝒃 𝟐 −𝒄 𝑝𝑞=𝑎−3𝑥 →𝒙= 𝒂−𝒑𝒒 𝟑
3−𝑝𝑥= 𝑎 →𝒙= 𝟑− 𝒂 𝟐 𝒑 1− 𝑥 =𝑦 →𝒙= 𝟏−𝒚 𝟐 𝑎 𝑏 2 = 𝑐 3 − 𝑥 →𝒙=± 𝒄 𝟑 −𝒂 𝒃 𝟐 𝑎− 𝑥 𝑏 =𝑐 →𝒙=𝒃 𝒂−𝒄 𝑎− 𝑥 𝑏−𝑑 =𝑐 →𝒙= 𝒂−𝒄 𝒃−𝒅 𝑎 𝑏−𝑥 =𝑑 →𝒙= 𝒂𝒃−𝒅 𝒂 3− 𝑥+𝑎 𝑏 =𝑐𝑑 →𝒙=𝒃 𝟑−𝒄𝒅 −𝒂 𝑞=3− 𝑐𝑑−𝑥 →𝒙=𝒄𝒅− 𝟑−𝒒 𝟐 ? 11 ? 2 12 ? 3 ? 4 ? ? 5 13 ? 6 ? ? 14 7 ? N ? 8 ? 9 ? N 10 ? ?

17. Vote with your diaries RED ORANGE GREEN BLUE
Test your understanding so far…
RED
ORANGE
GREEN
BLUE

18. 𝑦=3−𝑥
Make 𝒙 the subject.
x = y + 3
x = y – 3
x = 3 – y
x = 3y

19. 𝑦= 3 𝑥+1
Make 𝒙 the subject.
𝑥= 3 𝑦−1
𝑥= 3 𝑦 −1
𝑥= 𝑦 3 −1
𝑥= 𝑦 2

20. 𝑦=2 𝑥+𝑧
Make 𝒙 the subject.
𝑥= 𝑦 2 −𝑧
𝑥=𝑦−2𝑧
𝑥 = 𝑦−𝑧 2
𝑥= 2𝑦− 𝑧

21. 𝑦=1−2𝑥
Make 𝒙 the subject.
𝑥= 𝑦−1 2
𝑥 = 1+𝑦 2
𝑥=1−2𝑦
𝑥 = 1−𝑦 2

22. Skill #3: Subject trapped in a denominator
When the subject is in the numerator of a fraction, it’s easy to ‘release’ the subject from the fraction.
𝑦= 𝑥 𝑞
𝑥=𝑞𝑦 ?
But it’s a bit harder if the subject is in the denominator…
In general, whenever you have a fraction in an equation, your instinct should be to multiply both sides by the denominator.
𝑦= 𝑞 𝑥+1 ?
𝑦 𝑥+1 =𝑞
𝑥+1= 𝑞 𝑦 ?
𝑥= 𝑞 𝑦 −1 ?

23. Skill #3: Subject trapped in a denominator
! Isolate the fraction on one side of the equation, then multiply by denominator.
𝑎= 𝑏−𝑐 𝑥 𝒙= 𝒃−𝒄 𝒂
𝑎 𝑥 +𝑏=𝑐 𝒂 𝒙 =𝒄−𝒃 𝒂=𝒙 𝒄−𝒃 𝒂 𝒄−𝒃 =𝒙 1 2 ? ?
𝑎=𝑏− 𝑐 𝑥 𝒄 𝒙 =𝒃−𝒂 𝒙= 𝒄 𝒃−𝒂 3 ?

24. Doing it in one step… (if you like)
How would you rearrange the numbers in =3 to get another division?
12 3 =4
Thus we can swap the thing we’re dividing by and the result. The numerator is left unchanged. ?
Another way of thinking about it… Remember your speed-distance-time triangle from physics? Once you get to a point where you have to release 𝑥 from the fraction, you can apply what I call the ‘triangle trick’. ?
𝑐= 𝑏 4−2𝑥
4−2𝑥= 𝑏 𝑐
Examples:
𝑦= 𝑦 2 𝑥+𝑦 −2 𝒚+𝟐= 𝒚 𝟐 𝒙+𝒚 𝒙+𝒚= 𝒚 𝟐 𝒚+𝟐 𝒙= 𝒚 𝟐 𝒚+𝟐 −𝒚 E3
𝑎= 𝑏 𝑥 𝒙= 𝒃 𝒂 E1 E2
𝑎− 𝑏 2 𝑥+1 =𝑐 𝒙+𝟏= 𝒂− 𝒃 𝟐 𝒄
+2 first as was last thing done to 𝑥 ? ? ?

25. Click for Bromanimation
Skill #3b: ‘Cross multiplying’
If you have just a fraction on each side of the equation, you can ‘cross multiply’. 𝑎 𝑐 =
Click for Bromanimation 𝑏 𝑑
Examples:
Make 𝑥 the subject:
𝑏 2 𝑥−1 = 𝑐 𝑑
𝒃 𝟐 𝒅=𝒄 𝒙−𝟏 𝒃 𝟐 𝒅 𝒄 =𝒙−𝟏 𝒃 𝟐 𝒅 𝒄 +𝟏=𝒙 E2
𝑎+1 𝑏 = 𝑐 𝑥
𝒙 𝒂+𝟏 =𝒃𝒄 𝒙= 𝒃𝒄 𝒂+𝟏 E1 ? ?

26. Exercise 3
In each case make 𝑥 the subject of the formula. (Please copy out question first) ?
𝑎 𝑏 2 = 𝑐 𝑥 →𝒙= 𝒄 𝒂 𝒃 𝟐 𝑎= 𝑏 𝑥− →𝒙= 𝒃 𝒂 +𝟏 𝒐𝒓 𝒃+𝒂 𝒂 𝑎= 𝑏 2 𝑥 →𝒙= 𝒃 𝟐 𝒂−𝟏 1− 𝑐 2 𝑥 =𝑦 →𝒙= 𝒄 𝟐 𝟏−𝒚 𝑎 𝑥 =𝑝𝑞 →𝒙= 𝒂 𝒑𝒒 𝟐 𝑎+1 𝑥−1 +𝑐 𝑑 2 =𝑒 →𝒙= 𝒂+𝟏 𝒆−𝒄 𝒅 𝟐 +𝟏
𝑢𝑡 𝑢−𝑥 =𝑞 →𝒙=𝒖− 𝒖𝒕 𝒒
𝑟 2𝑥−3 = →𝒙= 𝒓+𝟏𝟐 𝟖 𝑎+1 2𝑥 = 𝑒 𝑓 →𝒙= 𝒇 𝟑 𝒂+𝟏 𝟐𝒆 ?
𝑦 1− 2 𝑥 =𝑐 →𝒙= 𝟐𝒚 𝒚−𝒄
𝑦+ 6 1+𝑥 =𝑦𝑞 →𝒙= 𝟔 𝒚𝒒−𝒚 −𝟏
𝑦= 1 𝑏−𝑎 𝑥 →𝒙=± 𝒃− 𝟏 𝒚 𝒂 𝒐𝒓 ± 𝒃𝒚−𝟏 𝒂𝒚
𝑞 2 𝑟=𝑞− 𝑞 3 𝑥−𝑞 →𝒙= 𝒒 𝟑 𝒒− 𝒒 𝟐 𝒓 +𝒒
𝑎 𝑏−2𝑥 = 2 𝑐 →𝒙= 𝟒𝒃− 𝒂 𝟐 𝒄 𝟐 𝟖
𝒚=1− 1 1− 1 1−𝑥 →𝒙=𝟏− 𝟏 𝟏 − 𝟏 𝟏−𝒚 𝟐 1 11 ? ? 2 12 ? 3 13 ? 4 ? ? 5 ? ? 6 14 ? ? 7 N ? N 8 ? ? 9

27. Questions similar to those in Tiffin CATs
Make 𝑢 the subject of the formula:
𝐷=𝑢𝑡+𝑘 𝑡 2
𝑢= 𝐷−𝑘 𝑡 2 𝑡 or 𝑢= 𝐷 𝑡 −𝑘𝑡 ?
Make 𝑑 the subject of the formula:
𝑟= 1 2 𝑐−𝑑 → 𝑑=𝑐−2𝑟 ?
Make 𝑒 the subject of the formula:
𝑝=2 𝑒+f → 𝑒= 𝑝 2 −𝑓 ?
Make 𝑥 the subject of the formula:
𝑞 2 = 𝑞 𝑠+𝑥 → 𝑥= 𝑞 𝑞 2 −1 −𝑠 ?

28. 𝑦= 𝑎 𝑥+1
Make 𝒙 the subject.
𝑥= 𝑦 𝑎−1
𝑥= 𝑎 𝑦 −1
𝑥= 𝑎 𝑦−1
𝑥=2

29. 𝑦= 𝑥 2 −1
Make 𝒙 the subject.
𝑥= 𝑦 2 +1
𝑥=2𝑦+1
𝑥=𝑦+2
𝑥=2𝑦+2

30. 𝑦=𝑎 𝑥 2
Make 𝒙 the subject.
𝑥=± 𝑎 𝑦
𝑥=± 𝑦 𝑎
𝑥= 𝑦 ± 𝑎
𝑥=± 𝑦𝑎

31. 𝑦= 2 𝑥 2
Make 𝒙 the subject.
𝑥=± 2 𝑦
𝑥= 2 𝑦
𝑥=± 𝑦 2
𝑥=± 2𝑦

32. 𝑦=𝑎 𝑥 −1 Make 𝒙 the subject. 𝑥= 𝑦 2 −1 𝑎 𝑥= 𝑦 2 +1 𝑎 𝑥= 𝑦 𝑎 +1 2
𝑦=𝑎 𝑥 −1
Make 𝒙 the subject.
𝑥= 𝑦 𝑎 +1 2
𝑥= 𝑦 2 −1 𝑎
𝑥= 𝑦 𝑎
𝑥= 𝑦 2 +1 𝑎

33. 𝑦= 1 𝑥+1 +1 Make 𝒙 the subject. 𝑥= 𝑦 2 −1 𝑥= 1 𝑦−1 2 −1 𝑥= 1 𝑦−1 2 −1
𝑦= 1 𝑥+1 +1
Make 𝒙 the subject.
𝑥= 1 𝑦−1 2 −1
𝑥= 1 𝑦−1 2 −1
𝑥= 1 1− 1−𝑦 2
𝑥= 𝑦 2 −1

34. 𝑎 𝑥 = 𝑏+1 𝑐 Make 𝒙 the subject. 𝑥= 𝑏𝑐 𝑎−1 2 𝑥= 𝑎𝑐 𝑏+1 2 𝑥=𝑎𝑏𝑐
𝑎 𝑥 = 𝑏+1 𝑐
Make 𝒙 the subject.
𝑥= 𝑏𝑐 𝑎−1 2
𝑥=𝑎𝑏𝑐
𝑥= 𝑎𝑐 𝑏+1 2
𝑥= 𝑎 𝑏+1 𝑐 2

35. Activity Time! There are 3 levels, each of increasing difficulty.
Once you’ve completed all the questions in Level 1, check your answers with me – after which you can advance onto the next level.
Merit on offer to anyone who can complete all Level 3 questions.

36. Substituting ? ? Q 𝒚=𝟒, 𝒂=𝟐𝟎 𝒄= 𝟐𝟎×𝟒 𝟏𝟐+𝟒 = 𝟖𝟎 𝟏𝟔 =𝟓
Bro Tip: Write out the values of your variables first using the information given.
𝒂=𝟑𝟎, 𝒄=𝟏𝟓 𝟏𝟓= 𝟑𝟎𝒚 𝟏𝟐+𝒚 𝟏𝟓 𝟏𝟐+𝒚 =𝟑𝟎𝒚 𝟏𝟖𝟎+𝟏𝟓𝒚=𝟑𝟎𝒚 𝟏𝟓𝒚=𝟏𝟖𝟎 𝒚=𝟏𝟐 ?
When we substitute our numbers in, we now have to to rearrange to solve!

37. Test Your Understanding
The maths exam mark 𝑚 of an 8EWS student is determined by the hours ℎ revised and the number of cats 𝑐 they have using the following formula:
𝒎= 𝟐𝒉+𝟑𝟎 𝒄
Given that Jaimal has 2 cats and gets a mark of 80, how many hours did he revise?
𝟖𝟎= 𝟐𝒉+𝟑𝟎 𝟐 𝟏𝟔𝟎=𝟐𝒉+𝟑𝟎 𝟏𝟑𝟎=𝟐𝒉 𝒉=𝟔𝟓
The ‘cool coefficient’ 𝐶 of a boy is determined by their number of skateboards 𝑠 and the mass ℎ of hair gel they apply, using the formula:
𝑪= 𝟏𝟎𝒉−𝟑𝒔 𝒉
a) If Max has a Cool Coefficient of 5 and has 10 skateboards, how many kilograms of hair gel does he use?
𝟓= 𝟏𝟎𝒉−𝟑𝟎 𝒉 𝟓𝒉=𝟏𝟎𝒉−𝟑𝟎 𝟓𝒉=𝟑𝟎 𝒉=𝟔
b) If he has a Cool Coefficient of 4 and uses 2kg of hair gel, how many skateboards does he have?
𝟒= 𝟐𝟎−𝟑𝒔 𝟐 𝟖=𝟐𝟎−𝟑𝒔 𝟑𝒔=𝟐𝟎−𝟖=𝟏𝟐 𝒔=𝟒 Q Q ? ? ?

38. Exercise 4
The price 𝑝 of a car with initial price 𝐶 and age 𝑎 is given by the formula:
𝑝=𝐶−3000 𝑎
Find the current price of the car when it is initially £ and is 5 years old. 𝒑=£𝟑𝟐𝟗𝟐
What is its age when a car initially worth £ has fallen to a tenth of its value? 𝒂=𝟗
You can calculate a temperature in Fahrenheit from Celsius using the formula:
𝐶= 5 9 𝐹−32
It is currently 30°𝐶. What is the temperature in Fahrenheit?
86F
𝑦= 1 1− 𝑥
Determine 𝑥 when 𝑦=5
𝟎.𝟔𝟒
𝑥=𝑎+ 𝑏𝑎 2 −𝑏 𝑏+𝑐
Find 𝑏 when 𝑎=2, 𝑐=3, 𝑥=4 𝒃=𝟔
Given that the conversion between Fahrenheit and Celsius is:
𝐶= 5 9 𝐹−32
Determine the temperature which is the same in both Fahrenheit and Celsius.
𝑭=𝑪 𝒔𝒐 𝑪= 𝟓 𝟗 𝑪−𝟑𝟐 Solving: 𝑪=−𝟒𝟎 4 6 1
𝑎= 𝑏+𝑐 𝑑
Given that 𝑎=5 and 𝑏=6 and 𝑑=7, find 𝑐
𝒄=𝟐𝟗
𝑎= 𝑏−𝑎 3𝑐
Given that 𝑐=1 and 𝑏=2, find 𝑎
𝒂= 𝟏 𝟐
𝑦=100−3 𝑥 2
Find 𝑦 when 𝑥=4 𝒚=𝟓𝟐
Find 𝑥 when 𝑦=−8 𝒙=𝟔 ? ? 7 ? 2 ? ? ? N 5 3 ? ? ? ?