Your first mate is a chemist (of course) so you have him mixing gunpowder back at the ship. He uses a mixture consisting of 14% potassium nitrate mineral ( x \rightarrow pounds) with a 24% potassium nitrate powder (y) to make 100 pounds of gunpowder containing 18% potassium nitrate.

Question

Which systems of linear equations models this situation?

Answer 1

To model this situation, we will set up a system of linear equations based on the information given. We are trying to mix two types of potassium nitrate to create a compound gunpowder.

Let:

• \( x \) = amount (in pounds) of the 14% potassium nitrate mineral.
• \( y \) = amount (in pounds) of the 24% potassium nitrate powder.

From the problem statement, we have two key pieces of information:

1. The total weight of the mixture should be 100 pounds: \[ x + y = 100 \]

2. The total amount of potassium nitrate in the mixture should provide 18% of the total weight (which is 100 pounds) as potassium nitrate. This means the total potassium nitrate from both components should equal 18 pounds (since \( 0.18 \times 100 = 18 \)):

   The amount of potassium nitrate contributed by the 14% mineral is \( 0.14x \) and from the 24% powder is \( 0.24y \). So, we can set up the second equation as follows: \[ 0.14x + 0.24y = 18 \]

Thus, the system of linear equations modeling this situation is: \[ \begin{cases} x + y = 100 \ 0.14x + 0.24y = 18 \end{cases} \]