Linear equations

[fishkeeper94]

How do I set up this linear equation? I have a 5 gallon tank that needs to be filled with 20% saltwater. I have 10% salt water that I can mix with 25% saltwater to get 20% ratio. How many gallons of 10% to 25% to equal 5 gallons of 20%? I only want to see how to set it up. Don't solve it please. Thank-you
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[DrLee]

How do I set up this linear equation? I have a 5 gallon tank that needs to be filled with 20% saltwater. I have 10% salt water that I can mix with 25% saltwater to get 20% ratio. How many gallons of 10% to 25% to equal 5 gallons of 20%? I only want to see how to set it up. Don't solve it please. Thank-you
:?:

Let X = the number of 10% seawater gallons needed and Y = the number of 25% seawater gallons needed.

We know right off the bat that X+Y=5 gallons.

But we also know that in that 5 gallons, there must be 20% salt or 0.20*5 units of salt. Therefore,
0.20*5 = 0.10*X + 0.25*Y.

From these two equations, you can find the solution ...

Let me know if that works ok ...

 

[soroban]

HGello, fishkeeper94!

I have a 5-gallon tank that needs to be filled with 20% saltwater.
I have a 10% saltwater solution that I can mix with a 25% saltwater solution.
How many gallons of each solution should I use?

\(\displaystyle \text{Let: }\;\begin{array}{ccc} x &=& \text{gallons of 10\% solution} \\ 5-x &=& \text{gallons of 25\% solution} \end{array}\)

\(\displaystyle \text{The }x\text{ gallons of 10\% solution contains: }\:0.10x\text{ gallons of salt.}\)

\(\displaystyle \text{The }5-x \text{ gallons of 25\% solution contains: }0.25(5-x)\text{ gallons of salt.}\)

. . \(\displaystyle \text{Together, the mixture contains: }\:0.10x + 0.25(5-x)\text{ gallons of salt.}\)


\(\displaystyle \text{But we know that the final mixture will be 5 gallons which is 20\% salt:}\)
. . \(\displaystyle \text{That is, it will contain: }\:0.20(5) \:=\:1\text{ gallon of salt.}\)


\(\displaystyle \text{We just described the final amoung of salt in }two\text{ ways.}\)

. . \(\displaystyle \text{There is our equation! }\quad \Rightarrow \quad 0.10x + 0.25(5-x) \;=\;1\)