Step 2: Choose variables to represent the unknown quantities. x amount of 8% solution needed y amount of 16% solution needed Step 3: Write a system of equations using two variables. Let’s begin by arranging the information in a table. Remember, to obtain the expression in the last column, multiply the percent of hydrogen peroxide in the solution by the amount of solution to get the amount of pure hydrogen peroxide in the solution. Mix these to make S Percent of Hydrogen Amount of Pure Peroxide in Solution Amount of Hydrogen Peroxide (as a decimal) Solution in Solution 0.08 x 0.08x 0.16 y 0.16y 0.10 200 0.10(200) To get one equation, use the information in the second column. It tells us that English: Amount of 8% solution plus Amount of 16% solution equals Amount of 10% solution T T T T T Equation: x y 200 The equation is x y 200. To get the second equation, use the information in the third column. It tells us that English: Amount of pure hydrogen peroxide in the 8% solution plus Amount of pure hydrogen peroxide in the 16% solution equals Amount of pure hydrogen peroxide in the 10% solution T T T T T Equation: 0.08x 0.16y 0.10(200) The equation is 0.08x 0.16y 0.10(200). The system of equations is x y 200 0.08x 0.16y 0.10(200). Step 4: Solve the system. Multiply the second equation by 100 to eliminate the decimals. Our system becomes x y 200 8x 16y 2000 Use the elimination method. Multiply the fi rst equation by 8 to eliminate x. 8x 8y 1600 8x 16y 2000 8y 400 y 50 Find x. Substitute y 50 into x y 200. x 50 200 x 150 The solution to the system is (150, 50). Remember that x and y must have a sum of 200, and neither should be negative. www.mhhe.com/messersmith SECTION 4.4 Applications of Systems of Two Equations 279
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