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When dealing with mixtures, concentration equations help us understand how a certain percentage of an ingredient (in this case, acid) is distributed in a solution. If you have a 10\(\%\) acid solution, it means that in every milliliter of solution, 10\(\%\) of it is pure acid. Similarly, for 20\(\%\) and 40\(\%\), you find that proportion of acid in the solutions. In our problem, we want a total of 18 mL of pure acid in a 100 mL solution. This is expressed mathematically as:- \(0.1x + 0.2y + 0.4z = 18\), where \(x\), \(y\), and \(z\) represent the milliliters of 10\(\%\), 20\(\%\), and 40\(\%\) acid solutions respectively. By setting up an equation like this, we lay the foundation for finding how much of each solution is needed.

A system of equations is a set of two or more equations that share variables. Solving a system means finding the values for these variables that satisfy all the equations in the system.In this problem:- We have three equations involving three unknowns. These represent different conditions: - Total volume: \(x + y + z = 100\) - Acid amount based on concentration: \(0.1x + 0.2y + 0.4z = 18\) - Relationship between 10\(\%\) and 40\(\%\) solutions: \(x = 4z\)By solving this system, we can determine exactly how many milliliters of each acid solution are needed to create the desired mixture.

Variables substitution is a powerful technique used to simplify solving systems of equations.In our exercise, we have:- \(x = 4z\) from the given condition about the ratios of solutions. This allows us to replace \(x\) with \(4z\) in our other equations. For instance:- The total volume equation becomes \(4z + y + z = 100\), simplifying to \(y + 5z = 100\).By substituting, we reduce the number of variables, making it easier to solve for the remaining ones.

Calculating the specific concentrations of acids in a final solution is the goal of our initial problem. Let's see how we apply our solved variables.We know:- \(x = 40\)- \(y = 50\)- \(z = 10\)Checking our conditions:- The first equation ensures the volumes sum up correctly: \(x + y + z = 40 + 50 + 10 = 100\).- The second checks acid concentration: \(0.1(40) + 0.2(50) + 0.4(10) = 4 + 10 + 4 = 18\).Thus, these calculations confirm that our solutions were placed in the correct amounts to achieve the desired concentration.