Try It! Use Linear Equations to Solve Mixture Problems 3. If the lab technician needs 30 liters of a \( 25 \% \) acid solution, how many liters of the \( 10 \% \) and the \( 30 \% \) acid solutions should she mix to get what she needs?

To solve this mixture problem, we can set up a system of linear equations. Let's define the variables: - Let \( x \) be the amount (in liters) of the \( 10\% \) acid solution. - Let \( y \) be the amount (in liters) of the \( 30\% \) acid solution. We know two things: 1. The total volume of the mixture should be \( 30 \) liters: \[ x + y = 30 \] 2. The total amount of acid in the mixture should equal \( 25\% \) of \( 30 \) liters, which is \( 0.25 \times 30 = 7.5 \) liters. The amount of acid from each solution can be expressed as: - From the \( 10\% \) solution: \( 0.10x \) - From the \( 30\% \) solution: \( 0.30y \) Therefore, we can write the second equation as: \[ 0.10x + 0.30y = 7.5 \] Now we have the following system of equations: 1. \( x + y = 30 \) 2. \( 0.10x + 0.30y = 7.5 \) We can solve this system step by step. ### Step 1: Solve the first equation for \( y \) From the first equation: \[ y = 30 - x \] ### Step 2: Substitute \( y \) in the second equation Substituting \( y \) in the second equation: \[ 0.10x + 0.30(30 - x) = 7.5 \] ### Step 3: Simplify and solve for \( x \) Expanding the equation: \[ 0.10x + 9 - 0.30x = 7.5 \] Combining like terms: \[ -0.20x + 9 = 7.5 \] Subtracting \( 9 \) from both sides: \[ -0.20x = 7.5 - 9 \] \[ -0.20x = -1.5 \] Dividing by \( -0.20 \): \[ x = \frac{-1.5}{-0.20} = 7.5 \] ### Step 4: Find \( y \) Now substitute \( x \) back into the equation for \( y \): \[ y = 30 - x = 30 - 7.5 = 22.5 \] ### Conclusion The lab technician should mix: - \( 7.5 \) liters of the \( 10\% \) acid solution - \( 22.5 \) liters of the \( 30\% \) acid solution Thus, the final answer is: - \( 7.5 \) liters of \( 10\% \) solution - \( 22.5 \) liters of \( 30\% \) solution