Some help with rearranging a formula

[taiko11]

Hi. I'm unsure if this is the right place to post, but i'm after some help. I have been out of school for a long time and I haven't done any math like this in about 15 years, so i'm very much behind the ball.

The equation I have been given is (listed in steps of rearranging)

1) 4.68 x10^-5 = ((2.5 - x)(5 - 2x)²) / (1 - y)

2) 4.68 x10^-5 = ((2.5 - x)(5 - 2x)²) / (1 - y) x10^-6

3) 46.8 (1 - y) = (2.5 - x) (4x² - 20x + 25)

4) 46.8 (x - 1) = -4x³ + 30x² - 75x + 62.5

5) 0 = -4x³ + 30x² – 121.8x + 109.3

So, I understand all of whats happening in the brackets. That's algebra I can remember. But I'm stumped at where the number x10^-6 is coming from at the end of (1-y) in step 2 and how they are changing 4.68 x10^-5 into 46.8 in step 3. Perhaps i've bitten off more than I can chew.

Any explanation would be appreciated. Thanks.

 

[mmm4444bot]

The equation I have been given is (listed in steps of rearranging)

1) 4.68 x10^-5 = ((2.5 - x)(5 - 2x)²) / (1 - y)

2) 4.68 x10^-5 = ((2.5 - x)(5 - 2x)²) / (1 - y) x 10^-6

3) 46.8 (1 - y) = (2.5 - x) (4x² - 20x + 25)

4) 46.8 (x - 1) = -4x³ + 30x² - 75x + 62.5

5) 0 = -4x³ + 30x² – 121.8x + 109.3

So, I understand all of whats happening in the brackets. That's algebra I can remember. But I'm stumped at where the number x10^-6 is coming from at the end of (1-y) in step 2 and how they are changing 4.68 x10^-5 into 46.8 in step 3.

Who gave you this list of steps? They are not correct.

Firstly, it's really bad form to use the letter x for two different things in the same exercise. It's true that we still write Scientific Notation using the letter x as a multiplication symbol, but not in equations containing the letter x as a variable.

Let's use an asterisk for a multiplication sign, instead.

1) 4.68 * 10^(-5) = [(2.5 - x)(5 - 2x)^2] / (1 - y)

Secondly, in line two, they have multiplied the right-hand side of the given equation by 10^(-6). If they do that, they would need to also multiply the left-hand side by the same amount. Either that's a typographical error, or they don't know what they've done.

Thirdly, in line 3, if they wanted to go from 4.68*10^(-5) to 46.8, then they need to multiply each side by 10^6, not 10^(-6).

Fourthly, in line 4, they have changed the factor (1-y) to (x-1). That's not justified, unless we're given a relationship between x and y (like: x+y=2, for example).

Finally, they have replaced the expression 46.8(x-1) with 0, and they also changed -75x to -121.8x and 62.5 to 109.3. These substitutions make no sense.

Please post the original exercise statement verbatim. Only then can everyone know that we're all on the same page. We can go from there. :cool:

 

[mmm4444bot]

If the original exercise is to solve the following equation for x, when y = 0, then the first step would be to substitute 0 for symbol y, and simplify.

4.68 * 10^(-5) = [(2.5 - x)(5 - 2x)^2]/(1 - y)

You could then multiply out the right-hand side, and, after that, subtract 62.5 from each side.

You'll then need a method to solve the resulting cubic equation. :cool:

 

[taiko11]

Sorry mate, didn't think about the x vs * thing. Ok, I was hoping I wasn't going to have to write this whole thing out.

This is a chemistry question on water treatment.

What will the composition of the treated water be when dosed with 2mmol/l lime?

1 mmol lime removes 2 mmol bicarbonate and 1 mmol calcium.
1 mmol lime also removes 1 mmol CO2.

so reaction ratio
a)1:2:1
b)1:1


Raw water composition

Calcium = 2.5 mmol/l
Bicarbonate = 5 mmol/l
CO2 = 1 mmol/l

Treated water composition (what will be left after treatment)

Calcium = x
Bicarbonate = 2x
CO2 = y



so, x + y = 2mmol

so y= 2 - x



The calcium carbonate equilibrium is

4.68 *10^-5 = ((2.5 - x)(5 - 2x)²) / (1 - y) (this is the original composition of the water 'minus' the treated water)

4.68 *10^-5 = ((2.5 - x)(5 - 2x)²) / (1 - y) *10^-6

46.8 (1 - y) = (2.5 - x) (4x² - 20x + 25)

46.8 (x - 1) = -4x³ + 30x² - 75x + 62.5

0 = -4x³ + 30x² – 121.8x + 109.3

Solve for x

x = 1.2 mmol/l y = 2 - 1.2 = 0.8 mmol/l



Hopefully that makes things clearer. It is possible that the 10*-6 is a typographical error and they meant 10*6.

Cheers.