## Monday, April 18, 2011

### Chapter 4: Algebra... Help! What's Wrong?

Given a and b are 2 unlike terms.
It is given that

• 3a + b
• 2s + 4t

Jane did the following algebraic manipulation:
• 3a + b = 3ab
• 2s + 4t = 6st
Do you think Jane is correct in her algebraic manipulations?
If yes, please write down examples to show that her answer is correct.

If not, explain to Jane her mistakes and help her to correct.
[you may substitute values for both a and  to prove your case]

1. Both answers are wrong as a+b is not ab as it cannot be simplified. Thus, there is no way of simplifying that answer.
For the next one, when we say 4t we cannot take the 4 as a number by itself. 4t is not 4 + t but it is t x 4. Also s+t is not st but remains the same as s+t. So this question cannot be simplified.

2. Both answers are wrong as only by multiplying a and b then will she get ab. The second one is also wrong. She should not add 2 and four together as these numbers show how many times that letter appears in the equation. So, it is actually like this: s+s+t+t+t+t=2s+4t. 2s+4t≠6st. She cannot solve these questions as they feature different numbers for the letters. She can only expand the equations, but not solve them.

3. Both of Jane's answers are wrong, as for:
a) The question stated 3a+b ( 3xa+b ), while 3ab ( Jane's Answer ) is 3xaxb, which is wrong.
i.e. a=2 b= 4.
3a+b=3x2+4
=10
3ab=3x2x4
=24
Thus, 3a+b≠3ab.
b) By Jane putting s and t together (st), it means that they are of the same terms, by are not, as stated in the question. 6st is 6xsxt, while 2s+4t is 2xs+4xt.
i.e. s=2 t=4
2s+4t=2x2+4x4
=20
6st=6x2x4
=48
Thus, 2s+4t≠6st.

4. Both of her answers are wrong as a+b is not ab.

i.e:
a = 5
b = 7

1. 3a + b = (3x5) + 7
= 22

2. 3ab = 3 x 5 x 7
= 105

The same concept applies for her next answer.

5. No Jane's answer is not correct in her algebraic manipulations, as "3a x b" only gives you the answer she stated, "3ab" and only "2s x 4t" equals to "6st". The correct answers for those questions are :

3a + b = (3a+b)
2s + 4t= (2s+4t)

6. Both of Jane's answers are wrong.
The expression a+b cannot be simplified. ab is only used when the expression is a*b. For the second question, she should not have added 2 and 4, she thought that 2 and 4 were separate numbers and added them. So, the second expression cannot be simplified any more.

7. This comment has been removed by the author.

8. Both of Jane's answers are wrong!!! [ Insert EVIL Laugh here! ]

AS.......

1 ) The expression a+b cannot be simplified.

ab is only used when the expression is ' 3 a*b '.

2 ) She should not have added 2 and 4,

as 2 and 4 are not constant but are coefficients of variables 's' and 't' respectively.

9. No.As a+b is already at its simplest form and it will be impossible lo simplifie it.If she wrote it how she wrote,the whole number stament and ans will be different.

10. Both of her answer are wrong, as a + b does not equals to ab, and s + t does not equals to st.

What she should have done:
a=2
b=3
3a + b = (2x3) + 3 = 9

s=5
t=7
2s + 4t = (5x2) + (7x4) = 10 + 28 = 38

11. No. It is incorrect. The reason is simply because of the fact that values a,b,s and t are different. If a is 3 and b is 4, 3x3=9 so if 3x3+4=13
But if the answer is 3ab, which is 21 is different Hence she is wrong

12. Both answers are wrong.3a+b does not give you 3ab and ,and 2s+4t does not give you 6st.She could have said:
A=5
B=10
3a+b=(3x5)+10=25

S=4
T=3
2s+4t=(2x4)+(4x3)=20

13. When you add 2 algebraic expressions, you cannot combine it. You can only combine algebraic expressions if the question asked for 'x' e.g AxB=AB
If it is A+B, the statement cannot be simplified further.

14. Both calculations are wrong.

First of all 'ab' = a x b ≠ a + b So we cannot assume that 3a + b = 3ab
(st ≠ s + t)
The next one has the same problem(2s + 4t ≠ 6st)
The other problem is that we cannot take 2s as 2 and 4t as 4 and add them together, so the equation of 2s+4t shouldn't be add together like ((2 + 4)+(s + t)).

15. Both answers are incorrect. 3a+b cannot be simplified into 3ab. 3ab is only used to show 3axb. 2s+4t cannot be simplified into 6st. 6st is only used to show 6sxt.

Let me show you an example for the first question.Let a be 3 and b be 4.
3a+b=3(3)+4
=9+4

Now, let me show you the wrong method which results to...
3a+b=3ab
=3(3)(4)
=3x3x4

Let me show you an example for the second question.Let s be 6 and t be 7.
2s+4t=2(6)+4(7)
=12+28
Now, let me show you the other wrong method Jane used which results to...
2s+4t=6st
=6(6)(7)
=6x6x7

16. Both of her answers are incorrect. As a and b have different value, they cannot be added together like what jane did. So for example(first question), take a as 5 while b as 6. What she is doing now is: 3(5)+(6)=3(5)(6) which is wrong. Same concept for the 2nd answer.

17. Both are wrong.

For the first statement, a and b cannot be simplified further to ab, therefore it is wrong.

For the second statement, both numbers 2 and 4 cannot be added as the unknown representing them are different.

18. Both answer are wrong as...

1) a and b are two different values and they cannot be simplify unless you multiply it.

2)a and b are different values and you cannot put them together(ab) only if you multiply it. 2 and 4 cannot add each other as the 2 must multiply a first then 4 must multiply b, then they can add each other.

19. Both are simply just - WRONG.

For the first statement, 3a + b = 3ab, is super wrong as a and b have separate numerical values and by grouping them as ab, it implies two things :

Either you are multiplying a AND b, as in the original staement, 2s + 4t

or,

ab is consisted of completely different numerical values from a and b.

For statement 2, 2s + 4t = 6st, like I said, s and t have completely different numerical values and putting it as st means that your either creating a new number or you are multiplying s and t. And what's more, she made a mistake as she wrote 6st. The st part was already wrong, and the numerical 2 and 4 are actually multiplied by either s or t. Simplified version:

Imagine s as 1 and t as 2.

So, in the original statement, it says: 2s + 4t. So, in the simplified version, it means 2 X 1 + 4 X 2, which equals to 10. But if I followed Jane's method (6st, or 6 X s X t), it would end up as 12 instead of 10. So that's what's wrong.

P.S. It was kinda expected that 'Jane" would make such a mistake :D but who knows ?