A chemical company makes two brands of antifreeze. The first brand is $55 \%$ pure antifreeze, and the second brand is $80 \%$ pure antifreeze. In order to obtain 80 gallons of a mixture that contains $75 \%$ pure antifreeze, how many gallons of each brand of antifreeze must be used? First brand: gallons Second brand: gallons

Step 1 ：A chemical company makes two brands of antifreeze. The first brand is 55% pure antifreeze, and the second brand is 80% pure antifreeze. The company wants to obtain 80 gallons of a mixture that contains 75% pure antifreeze. We need to find out how many gallons of each brand of antifreeze must be used.

Step 2 ：We denote the volume of the first brand used as \(x\) and the volume of the second brand used as \(y\).

Step 3 ：We can set up a system of linear equations to represent this problem. The first equation will represent the total volume of the mixture, and the second equation will represent the total amount of pure antifreeze in the mixture.

Step 4 ：The first equation is \(x + y = 80\), since the total volume of the mixture is 80 gallons.

Step 5 ：The second equation is \(0.55x + 0.8y = 0.75 * 80\), since the total amount of pure antifreeze in the mixture is 75% of the total volume.

Step 6 ：Solving this system of equations, we find that \(x = 16\) and \(y = 64\).

Step 7 ：Final Answer: The company must use \(\boxed{16}\) gallons of the first brand and \(\boxed{64}\) gallons of the second brand to obtain 80 gallons of a mixture that contains 75% pure antifreeze.