Solve Basic Algebra Equations & Find Solution Amounts | 3 Significant Figures

[FaraDazed]

Homework Statement

Question A: Solve the following equation for x, express answer in standard form to three significant figures.

$$24 = \frac{3x-1}{7x}-\frac{2}{3x}$$

Question B:
A solution containing 13% alcohol is then mixed with a another solution containing 5%
alcohol. When they are combined there is a total of 24 litres of solution containing 8% alco-
hol.

Write down two equations that represent the information given, and then using alegbra, find out how much of each solution is present.

Homework Equations

n/a

The Attempt at a Solution

I think I have done question A. There is probably a better/easier way of doing it but hey,
$$24 = \frac{3x-1}{7x}-\frac{2}{3x} \\ 24 = \frac{3x-1}{7x}-\frac{2\frac{7}{3}}{3x\frac{7}{3}} \\ 24 = \frac{3x-1}{7x}-\frac{\frac{14}{3}}{7x} \\ 24 = \frac{3x-1-\frac{14}{3}}{7x} \\ 168x=3x-1-\frac{14}{3} \\ 165x=-\frac{17}{3} \\ x=\frac{-\frac{17}{3}}{165}=-0.0343 = 3.43 ×10^{-2}$$

Question B I am really stuck on. At the moment I have just tried to write down the equations, probably completely wrong, but its all I have.

$$0.13A_1+0.05A_2=0.08(24)$$
Where A is the amount of each solution but have no idea if that is correct and/or how to progress

 

[phyzguy]

You've done Question A correctly except you dropped the minus sign in the final answer.

For Question B, the equation you have written is correct, but you need to write a second equation connect A1 and A2. Try writing an equation expressing the total amount of liquid.

Then once you have two equations for A1 and A2, you should be able to solve them.

 

[FaraDazed]

Thanks for your reply :)

When you said B1, did you mean the A2 in the equation I put or the 24 coefficiant on the right hand side of the equation?

Also, you said "Then once you have two equations for A1 and A2" did you mean two equations in total or two equations for each A1 and A2, meaning four in total? If it is the former, then is it simply a case of rearanging the equation I have already or is it a new equation I need to do, if it is that I just can't see it, really stuck.

 

[phyzguy]

Thanks for your reply :)

When you said B1, did you mean the A2 in the equation I put or the 24 coefficiant on the right hand side of the equation?

Also, you said "Then once you have two equations for A1 and A2" did you mean two equations in total or two equations for each A1 and A2, meaning four in total? If it is the former, then is it simply a case of rearanging the equation I have already or is it a new equation I need to do, if it is that I just can't see it, really stuck.

Sorry, I meant to say that you need to write a second equation connecting A1 and A2 I corrected the post.

You have two unknowns, A1 and A2, so you need two equations in total to be able to solve for the two unknowns. You have one, so you need a second one. No, you can't simply re-arrange the equation you have, because that will not give you any new information. You need a second equation independent of the first. The one you wrote expresses that the sum of the amount of alcohol in the two solutions equals the amount of alcohol in the final mixture. Try writing a similar equation that expresses that the sum of the total amount of liquid in the two solutions equals the total amount of liquid in the final mixture.

 

[FaraDazed]

Sorry, I meant to say that you need to write a second equation connecting A1 and A2 I corrected the post.

You have two unknowns, A1 and A2, so you need two equations in total to be able to solve for the two unknowns. You have one, so you need a second one. No, you can't simply re-arrange the equation you have, because that will not give you any new information. You need a second equation independent of the first. The one you wrote expresses that the sum of the amount of alcohol in the two solutions equals the amount of alcohol in the final mixture. Try writing a similar equation that expresses that the sum of the total amount of liquid in the two solutions equals the total amount of liquid in the final mixture.

OK thanks, is it as simple as A1+A2=24? If it is that it doesn't surprise me as I always miss the simple things.

 

[FaraDazed]

OK I think I have done it. Using A1+A2=24 and rearranging to get A1=24-A2 and that A2=24-A1 and then subbing them into the original equation I had.

$$0.13A_1+0.05A_2=0.08(24) \\ 0.13(24-A_2)+0.05A_2=1.92 \\ 3.12-0.13A_2+0.05A_2=1.92 \\ -0.13A_2+0.05A_2=-1.2 \\ -0.08A_2=-1.2 \\ A_2=\frac{1.2}{0.08}=15$$

Then I did the same to find A1 and got 9, and 15+9 = 24 so I think I have done it correct.

Thanks for your help :)

 

[phyzguy]

Looks good. Glad to help.