linear systems word problem

Question

Steve is make a lemonade, he needs to make 56 cups of lemonade. The recipe requires 2 cups of lemon juice for every 5 cups of water added. How many cups of lemon juices are required?

I did:

Le5 x be number of cups of lemon juice, y be the number of cups of water $$2x+5y=56$$

But this isn't correct. What would be the correct equation?? And also I am pretty sure I meed 2 equations.

Answer 1

Let´s assume that the sizes of the cups are all equal. And one cup lemonade requires two parts lemon juice and 5 parts water. In one cup lemonade there are 7 parts. Two of the 7 parts are lemon juice. Therefore in one cup lemonade there are $\frac27 $ cups lemon juice.

Thus you need $\frac27 \cdot 56=16$ cups lemon juice to get 56 cups lemonade.

Answer 2

You can get one equation by observing, that mixing $A$ cups of water and $B$ cups of lemon juice, if gives you $A+B$ cups of lemonade. You want 56 cups of lemonade.

A second equation is given by tha ratio of 2 to 5. Since there is so much more water, you need to compare five times of the cups of $A$ with 2 times of the cups of $B$.

I think you can go on from here.

Answer 3

I have found a different solution after couple days,

let $x$ represent the number of cups of lemon juice, and $y$ represent the number of cups of water

$$x+y=56 ---(1)\\ {x \over y}={2 \over 5}-----(2)$$ $$(2): x= {2y \over 5}$$ Substitute (2) into (1): $${2y \over 5} +y = 56 \\ 2y +5y = 280\\ 7y = 280 \\ y =40$$ Substitute $y=40$ into (1): $$x+40 = 56 \\ x=16$$

$\therefore$ 16 cups of lemon juice is required