MHB Solve Word Problem w/ Matrices: Chapter I

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The discussion revolves around solving a word problem involving the mixing of two solutions using matrices. The problem requires finding the amounts of a 60% solution and a 20% solution needed to create 100 liters of a 50% solution. Two equations are established: one for the total volume of the mixture (x + y = 100) and another for the active ingredient (0.6x + 0.2y = 50), which is simplified to 3x + y = 250 for easier matrix representation. The matrix equation is set up as [1 1; 3 1][x; y] = [100; 250]. The simplification of coefficients to integers by multiplying by 5 is noted as a helpful step in the solution process.
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I have a word problem that I am struggling with. I have been using matrices in this chapter, but I don't understand how it applies or where to start in order to solve this equation. Here is the word problem:

One hundred liters of a 50% solution is obtained by mixing a 60% solution with a 20% solution. How many liters of each solution must be used to obtain the desired mixture?
 
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I would let $x$ represent the amount (in liters) of the 60% solution needed, and $y$ be the number of liters of the 20% solution required. Since the final desired outcome is 100 liters of solution, we know:

$$x+y=100$$

We also know that we will need in the final solution 50L of the active ingredient, $0.6x$ coming from the 60% solution and $0.2y$ coming from the 20% solution, then we also have:

$$0.6x+0.2y=50$$

or:

$$3x+y=250$$

So, we can set up our matrix equation as follows:

$$\left[\begin{array}{c}1 & 1 \\ 3 & 1 \end{array}\right]\left[\begin{array}{c}x \\ y \end{array}\right]=\left[\begin{array}{c}100 \\ 250 \end{array}\right]$$

Can you proceed?
 
I can proceed. But I don't understand how .6x and .2y became 3 and 1.

MarkFL said:
I would let $x$ represent the amount (in liters) of the 60% solution needed, and $y$ be the number of liters of the 20% solution required. Since the final desired outcome is 100 liters of solution, we know:

$$x+y=100$$

We also know that we will need in the final solution 50L of the active ingredient, $0.6x$ coming from the 60% solution and $0.2y$ coming from the 20% solution, then we also have:

$$0.6x+0.2y=50$$

or:

$$3x+y=250$$

So, we can set up our matrix equation as follows:

$$\left[\begin{array}{c}1 & 1 \\ 3 & 1 \end{array}\right]\left[\begin{array}{c}x \\ y \end{array}\right]=\left[\begin{array}{c}100 \\ 250 \end{array}\right]$$

Can you proceed?
 
megacat8921 said:
I can proceed. But I don't understand how .6x and .2y became 3 and 1.

I multiplied the equation by 5 so that all coefficients are integers. :D
 
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