any maths fans there? can you help me please?

Question

Please would you help me with these questions. I don't mind if you can do all of it, some of it or just one-any amount of help will be really appreciated. i promise

1.In this question, a and b are numbers where a=b + 2
The sum of a and b is equal to the product of a and b
show that a and b are not integers

2.show that (4x10^8) x (8x10^4) = 3.2 x 10^13

3.what is (4x10^8) / (8x10^4) write answer in standard form

4.Here is information about a data set.
"There are 100 values in the set
The median is 90
The mean is 95"
i increase the highest value in the data set by 200. Now what are the median and the mean of the data set?

5.solve these simultaneous equations using an algebrating method
4x + 3y =21
2x + y = 8
show working


thanks, it's a bit of holiday homework, it was a whole 50 question booklet, i only missed out these so i really appreciate if you help me on just these few.however if you think it is wrong helping me then please don't be cruel.

Answer 1

I'll do the last one for ya cause I'm a bit bored.

multiply the second equation by two to give

4x+3y=21
4x+2y=16

Take bottom from top

4x+3y=21
- 4x+2y=16

= y=5

substitute y=5 into either of the above equations to find x

Answer 2

(1)
a = b + 2
ab = a + b

Plug in the first into the second:

ab = a + b
(b + 2)b = (b + 2) + b
b^2 + 2b = 2b + 2
b^2 = 2

You can't square an integer to get 2. The two solutions are:

b = sqrt(2), a = 2 + sqrt(2)
b = -sqrt(2), a = 2 - sqrt(2)

==============
(2) 4 x 10^8 * 8 x 10^4 =
(4 * 8) x (10^(4 + 8)) =
32 x 10^12

However, that's not proper form for scientific notation, since there are two digits to the left of the decimal point. One more factor of 10 has to be moved into the exponent:

32 x 10^12 =
3.2 * 10 * 10^12 =
3.2 x 10^13

==============
(3) (4 x 10^8) / (8 x 10^4) =
(4/8) x 10^(8-4) =
0.5 x 10^4

Again, not proper notation, so we have to move a factor of 10 out of the exponent:

0.5 x 10^4 =
(0.5 * 10) x 10^3 =
5 x 10^3

==============
(4) The median is the middle value of the data set. Making the largest value even larger will not change which one is the middle value, or how much that value is, so the median does not change. It is still 90.

The mean was of the form (where "s" is the sum of all the data):

s/100 = 95

This tells us that s=9,500

If you add 200 to one number, the new sum would be 9,500 + 200, which is 9,700. And that means the new mean would be:

9,700/100 = 97

==============
(5)
4x + 3y =21
2x + y = 8

Second equation (solved for y) is y = 8 - 2x, plug that into the first:

4x + 3y = 21
4x + 3(8 - 2x) = 21
4x + 24 - 6x = 21
2x = 3
x = 3/2

Use the now-known value for x and either original equation to solve for y:

2x + y = 8
2(3/2) + y = 8
3 + y = 8
y = 5

Check using the other equation:

4x + 3y =? 21
4(3/2) + 3(5) =? 21
6 + 15 =? 21
21 = 21

Answer 3

If you can do 3, then 2 is obvious.
3) (4x10^8) / (8x10^4) = (4/8) x (10^8/10^4)
= 0.5 x 10^(8-4)
= 5 x 10^(-1) x 10^4
= 5 x 10^3
2) LHS = (4x10^8) x (8x10^4)
= (4 x 8) x (10^8 x 10^4)
= 32 x 10^(8+4)
= 3.2 x 10^1 x 10^12
= 3.2 x 10^13
Thus, LHS = RHS. Shown.

4) Median will not changed, because it is the middle value. The highest value increased by 200 is still the highest value. Thus, obviously it did not become a value lower than the median, thus 90 is still the median. Anyway, if median did change, the question does not have sufficient information for you to know the new median value.
Mean will be changed. Mean = sum of all values / number of values. Since the sum of all values increased by 200 while the number of values remains at 100, mean is increased by 200/100 = 2. Thus, the mean is 97 now.

Answer 4

1.

a = b+2
ab = a+b

now substitute b+2 for a

(b+2)b = b+2 + b
b² + 2b = 2b + 2
b² = 2
b = ±√2

which is clearly not an integer.

====

2.

(4x10^8) x (8x10^4)
4x8 x 10^4x10^8
32x10^(8+4)
3.2 x 10^1 x 10^12
3.2 x 10^(12+1)
3.2 x 10^13

====

3.

(4x10^8)/(8x10^4)
(4 / 8) x (10^8 / 10^4)
0.5 x 10^(8-4)
5x10^-1 x 10^4
5x10^(-1 + 4)
5x10^3

====

4.

The meadian is where the middle of the distribution lies. Increasing the largest number doesn't alter the distribution so the median remains the same.

The mean is the average of each element which is calculated as the sum of all elements divided by the number of elements. Since there are 100 elements in the set, increasing one member by 200 increases the sum of all the elements in the set by 200 which is 200/100 or 2 per element. Thus the mean increases by 2 per element from 95 to 97.

====

5.

4x + 3y = 21
2x + y = 8

multiply the second equation by 2

2x + y = 8 ==> 4x + 2y = 16

Now subtract this from the first equation

4x + 3y = 21
4x + 2y = 16
-----------------
y = 5

Replace y with 5 in the second equation

2x + 5 = 8
2x = 3
x = 3/2

Answer 5

Ok I'm rusty at this but what the heck, I'll try a couple...
1) if a=b+2 and a+b=a*b then by sub. you should end up with 2b+2=b^2+2 or 2b=b^2 this can not be an integer.
5) solve the second equation for y giving y = -2x+8 then sub this into the first giving 4x+3(-2x+8)=21 solving down to x=3/2; then plug this into the second equation to solve for y=5

Answer 6

1)
a+b=ab
a=b+2
so b+2+b=(b+2)b
2b+2=b^2+2b
b^2=2
b=+/-root 2
a=+/-root 2 +2
root 2 is an irrational number, a and b belong to the irrational, not contained in I

2)
(4*10^8)(8*10^4)
=4*8*10^12
=32*10^12
=3.2*10^13

3)
(4*10^8)/(8*10^4)
=(4*10^4)/8
= 4/8*10^4
=.5*10^4
=5.0*10^3

4)
(99*95+295)/100
=97 (new mean)
median is unchanged

5)
4x+3y=21
2x+y=8
4x+3y-4x-2y=21-16
y=21-16
y=5
2x+5=8
x=3/2

Answer 7

#1. a = b + 2
a + b = ab
b + 2 + b = (b + 2) b
2b + 2 = b^2 + 2b
2 = b^2
b = sqrt (2)
sqrt(2) is not an integer.
Therefore, a and b or not integers

#5.
(1) 4x + 3 y = 21
(2) 2x + y = 8
(3) (double (2)) 4x + 2y = 16
(4) (subtract (1) - (3)) 0x + 1y = 21 - 16 = 5
(5) (simplify (4)) y = 5
(6) (plug y into (2)) 2x + 5 = 8
(7) 2x = 3
(8) x = 3 /2
Check with (1)
4(3/2) + 3(5) =? 21
12/2 + 15 =? 21
6 + 15 =? 21
21 =? 21, yes.

Answer 8

1) a+b = a(b) ... a= b+2
--- b+2 + b = (b+2)b
--- 2b + 2 = b^2 +2b
--- 2 = b^2
--- b = sqrt 2
--- a = b+2
--- a = (sqrt 2) + 2

2) (4x10^8) (8x10^4) = 4(8) x10^(8+4)
---- = 32 x10^12 or 3.2 x10^13

3) (4x10^8) / (8x10^4) = (4/8) x10^(8-4)
---- = 0.5 x10^4
---- = 5 x10^3 or 5000

5) 4x + 3y = 21 ----> y = (21-4x)/3
-- 2x + y = 8 ------> y = (8 - 2x)

equate: (21-4x)/3 = 8-2x
--- 21-4x = 3(8-2x)
--- 21-4x = 24 - 6x
--- 2x = 3
--- x = 3/2

2x + y = 8
2(3/2) + y = 8
3 + y = 8
y = 5

Answer 9

for question 5)

equation A) 4x+3y=21
equation B)2x+y=8

so multiply equation B by 2

equation C) 4x+2y=16

take equation C from A

4x+3y=21
4x+2y=16
=0x+y=5
therefore
y=5

now use y as 5

4x+(3*5)=21
4x+15=21
4x=21-15
4x=6
x=6/4
x=1.5

check with equation B
(2*1.5)+5
3+5=8

so x=1.5 and y=5

hope this helps you and that you understand my workings out!

Answer 10

The 1st ones a bit easy...

a=b+2

a+b=a*b, if we substitute a=b+2, the answer is
b=2^1/2

if we substitue b to a=b+2, the answer is 2+2^1/2