Simultaneous equation: 3 * ([x * 71 / 40] + 6.5) = [y * 96 / 40] + 0.9

coupe72001

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Hi,
I need the working for a partly solved simultaneous equation:

[FONT=&quot]3 * ([x * 71 / 40] + 6.5) = [y * 96 / 40] + 0.9

whereby

[/FONT]
[FONT=&quot]you can substitute (88.7 – x) for y and get x = 25.1
and
once you have x, you can solve again and get y = 63.6

Clearly the order of operation that I am using is wrong, as I don't get the correct answers supplied when I attempt this!
Thanks in advance.[/FONT]
 
Hi,
I need the working for a partly solved simultaneous equation:

3 * ([x * 71 / 40] + 6.5) = [y * 96 / 40] + 0.9

whereby

you can substitute (88.7 – x) for y and get x = 25.1
and
once you have x, you can solve again and get y = 63.6

Clearly the order of operation that I am using is wrong, as I don't get the correct answers supplied when I attempt this!
Thanks in advance.
Can you share ALL your work - so that we can understand your mistake and unravel it.
 
Hi,
I need the working for a partly solved simultaneous equation:

[FONT=&quot]3 * ([x * 71 / 40] + 6.5) = [y * 96 / 40] + 0.9

whereby

[/FONT]
[FONT=&quot]you can substitute (88.7 – x) for y

You can? Do you mean that the equation above is one of a pair of simultaneous equations and y= 88.7- x is the other?

Okay, if y= 88.7- x the original equation becomes 213x/40+ 19.5= 96y/40- 0.9= 96(88.7- x)/40- 0.9
Multiply through by 40 to get 213x+ 780= 96(88.7- x)- 36= 8515.2- 96x- 36= 8479.2- 96x.

Add 96x to both sides and subtract 780 from both sides to get
409x= 8699.2.
Divide both sides by 409: x= 8699.2/409= 88.269437652811735941320293398533 which rounds to 88.3, to one decimal place, not 25.1

and get x = 25.1
and
once you have x, you can solve again and get y = 63.6

Clearly the order of operation that I am using is wrong, as I don't get the correct answers supplied when I attempt this!
Thanks in advance.[/FONT]
 
Sorry, bit of background for clarity... or confusion. Here is the full question, and the answer supplied by the course co-ordinator in red. There was also a table supplied to show how much Ca, Cl2 and SO4 are in the Melbourne and London water profiles in ppm, and the idea was to find a way to increase Ca from 1.3 to 90ppm, and have Cl2 (molecular weight 71) and SO4 (molecular weight 96) in a 3:1 ratio. This changes one areas water profile to the other by the mixing two salts (using the known molecular weights, and with the starting values of 1.3 ppm Ca, 6.5ppm Cl2 and 0.9 SO4 already present in Melbourne water). It's a beer brewing problem.

Question:
Your water composition is the same as Melbourne, and Portersare traditionally made from “London” water. For this exercise, you want to increasethe calcium to 90 ppm (from 1.3ppm) and have a ratio of sulphate to chloride of 3:1. You haveaccess to the following salts:

CaCl2.2H2O Calcium Chloride
CaSO4.2H2O Calcium Sulphate / Gypsum

How much calcium chloride and gypsum will you add per 1000 L of water?

Answer:
Let x = ppm Ca from CaCl2.2H2O
Let y = ppm Ca from CaSO4.2H2O

You want x + y to equal 90, less the Ca in Melbourne water, so x + y = 88.7 and y = 88.7 – x

So for every bit of Ca you add from CaCl2.2H2O, you add 71/40 of Cl (2xCl, so 2 x 35.5)
So for every bit of Ca you add from CaSO4.2H2O, you add 96/40 of SO4 (1 x 32, + 4 x 16)

The ppm Cl– added = x * 71/40
The ppm SO42– added = y * 96/40
You want sulphate to chloride = 3
You already have 6.5 ppm Cl and 0.9 ppm SO4
Therefore: 3 * ([x * 71 / 40] + 6.5) = [y * 96 / 40] + 0.9

Substitute (88.7 – x) for y in above, solve the equation, and I get x = 25.1 ppm Ca as CaCl2.2H2O and y = 63.6ppm Ca as CaSO4.2H2O.

Calculate grams of salts (eg 25.1 x 147/40 for CaCl2.2H2O), and get 92.4 and 273.3ppm as CaCl2.2H2O and CaSO4.2H2O respectively. The units ppm (mg/L) are the same as g / 1000L.

While I understand the overarching principles here, I always bomb out with the maths when attempting to solve 3* ([x * 71 / 40] + 6.5) = [y * 96 / 40] + 0.9
(whereby x = 25.1 and y = 63.6 as per the feedback from the course co-ordinator)

I've been beginning with the substitution for y like this:
3* ([x * 71/40] + 6.5) = [88.7-x * 96/40] + 0.9

and trying to finish with the given x=25.1

However, I'm not sure what the first step in the order of operations is, or what to do with the x * fraction portion of the problem. For x * 71/40 I've been solving as 1.775x. Clearly that's wrong.
 
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Many years ago, they introduced "individual synchronized swimming" in the summer Olympics. My thought was "how can one person be synchronized?" How can one equation be "simultaneous"?
 
True. Simplifying leads to: 32y - 71x = 248
Now do the substitutions...

Proof:
https://www.wolframalpha.com/input/?i=3+*+((x+*+71+/+40)+++6.5)+=+(y+*+96+/+40)+++0.9,+y=88.7-x

I'm glad we have confirmed the supplied answer corresponds with the supplied formula, and it's only my maths that's off.
I don't suppose anyone would share the step-by-step solution from wolframalpha? I'd rather not get a monthly subscription for a single problem.
I'm not reaching 32y-71x = 248 or understand how to solve 32(88.7-x) - 71x = 248.

This is how far I get:

3 * ((x * 71 / 40) + 6.5) = (y * 96 / 40) + 0.9, y=88.7-x

Step 1. Multiply by 3 first to get

213x/40 + 19.5 = (y* 96/40) + 0.9

Step 2. Multiply both sides by 40 to get

213x + 780 = 96y + 36

Step 3. Subtract 36 from both sides

213x + 744 = 96y

Step 4. Subtract 213x from both sides

744 = 96y - 213x


... so I'm way off the correct answer 32y - 71x = 248 and not sure how to complete the substitution anyway.
 
I'm glad we have confirmed the supplied answer corresponds with the supplied formula, and it's only my maths that's off.
I don't suppose anyone would share the step-by-step solution from wolframalpha? I'd rather not get a monthly subscription for a single problem.
I'm not reaching 32y-71x = 248 or understand how to solve 32(88.7-x) - 71x = 248.

This is how far I get:

3 * ((x * 71 / 40) + 6.5) = (y * 96 / 40) + 0.9, y=88.7-x

Step 1. Multiply by 3 first to get

213x/40 + 19.5 = (y* 96/40) + 0.9

Step 2. Multiply both sides by 40 to get

213x + 780 = 96y + 36

Step 3. Subtract 36 from both sides

213x + 744 = 96y

Step 4. Subtract 213x from both sides

744 = 96y - 213x


... so I'm way off the correct answer 32y - 71x = 248 and not sure how to complete the substitution anyway.


Anyone keen to revisit this? We have the question and the answer, but need some guidance on where my steps in the working is wrong, and how to complete the substitution part.
Pretty close to just paying for wolframalpha to just spell it out.

Cheers
 
Anyone keen to revisit this? We have the question and the answer, but need some guidance on where my steps in the working is wrong, and how to complete the substitution part.
Pretty close to just paying for wolframalpha to just spell it out.

Cheers
\(\displaystyle 3 * \left \{ \left( x * \dfrac{71}{40} \right ) + 6.5 \right \} = \left ( y * \dfrac{96}{40} \right ) + 0.9 \implies\)

\(\displaystyle \dfrac{213x}{40} + 19.5 = \dfrac{96y}{40} + 0.9 \implies\)

\(\displaystyle 40 * \left ( \dfrac{213x}{40} + 19.5 \right ) = 40 * \left ( \dfrac{96y}{40} + 0.9 \right) \implies\)

\(\displaystyle 213x + 780 = 96y + 36 \implies\)

\(\displaystyle 213x + 744 = 96y.\)

\(\displaystyle \text {And } y = 88.7 - x.\)

\(\displaystyle \therefore 213x + 744 = 96(88.7 - x) \implies\)

\(\displaystyle 213x + 744 = 8512.1 - 96x \implies\)

\(\displaystyle 213x + 96x = 8512.2 - 744 \implies\)

\(\displaystyle 309x = 7768.2 \implies\)

\(\displaystyle x = \dfrac{7768.2}{309} \approx 25.1398.\)

\(\displaystyle \therefore y \approx 88.7000 - 25.1398 = 63.5602.\)

As you can see, I do not get wolfram's answer exactly. Probably I transposed a digit or something in putting things into my calculator. But this is the process.
 
\(\displaystyle 3 * \left \{ \left( x * \dfrac{71}{40} \right ) + 6.5 \right \} = \left ( y * \dfrac{96}{40} \right ) + 0.9 \implies\)

\(\displaystyle \dfrac{213x}{40} + 19.5 = \dfrac{96y}{40} + 0.9 \implies\)

\(\displaystyle 40 * \left ( \dfrac{213x}{40} + 19.5 \right ) = 40 * \left ( \dfrac{96y}{40} + 0.9 \right) \implies\)

\(\displaystyle 213x + 780 = 96y + 36 \implies\)

\(\displaystyle 213x + 744 = 96y.\)

\(\displaystyle \text {And } y = 88.7 - x.\)

\(\displaystyle \therefore 213x + 744 = 96(88.7 - x) \implies\)

\(\displaystyle 213x + 744 = 8512.1 - 96x \implies\)

\(\displaystyle 213x + 96x = 8512.2 - 744 \implies\)

\(\displaystyle 309x = 7768.2 \implies\)

\(\displaystyle x = \dfrac{7768.2}{309} \approx 25.1398.\)

\(\displaystyle \therefore y \approx 88.7000 - 25.1398 = 63.5602.\)

As you can see, I do not get wolfram's answer exactly. Probably I transposed a digit or something in putting things into my calculator. But this is the process.


Brilliant!
Please note I am definitely kicking myself for not picking I was exactly threefold the correct answer.
Great to see the substitution work as well.
I'll have to use this process over and over (albeit with different values) so having a absolutely clear step-by step process is vital.
Thanks all who contributed to this.
 
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