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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 75
Chapter 6, Problem 75

Use a system of linear equations to solve Exercises 73–84. How many ounces of a 15% alcohol solution must be mixed with 4 ounces of a 20% alcohol solution to make a 17% alcohol solution?

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Define the variable: Let x represent the number of ounces of the 15% alcohol solution to be mixed.
Set up the equation based on the total amount of alcohol in the mixture: The amount of alcohol from the 15% solution is 0.15⋅x, and from the 20% solution is 0.20⋅4.
Express the total amount of alcohol in the final mixture, which has a concentration of 17% and a total volume of x + 4 ounces, as 0.17⋅(x + 4).
Write the equation representing the balance of alcohol content: 0.15x + 0.20 imes 4 = 0.17(x + 4).
Solve the equation for x by first expanding the right side, then collecting like terms, and finally isolating x to find the number of ounces of the 15% solution needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. In mixture problems, these equations represent relationships between quantities and concentrations.
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Mixture Problems and Concentration

Mixture problems involve combining substances with different concentrations to achieve a desired concentration. The key is to set up equations based on the total amount and the amount of the substance of interest (e.g., alcohol) in each part of the mixture.
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Setting Up and Solving Equations from Word Problems

Translating a word problem into equations requires identifying variables, writing expressions for quantities and concentrations, and forming equations that represent the problem conditions. Accurate setup is crucial for solving the problem correctly.
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In Exercises 69–70, rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. |x|≤2, |y|≤3

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In Exercises 65–68, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the x-variable and the y-variable is no more than 2. The y-variabe is no less than the difference between the square of the x-variable and 4.

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Textbook Question

Use a system of linear equations to solve Exercises 73–84. How many ounces of a 50% alcohol solution must be mixed with 80 ounces of a 20% alcohol solution to make a 40% alcohol solution?

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Textbook Question

Solve the systems in Exercises 79–80.

{logy x=3logy (4x)=5\(\left\)\{\(\begin{array}{l}\]\log\)_{y}\(\text{ x=3}\)\\ \(\log\)_{y}\(\text{ (4x)=5}\[\end{array}\]\right\).

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Textbook Question

Solve the systems in Exercises 79–80.

{log x2=y+3log x=y−1\(\left\)\{\(\begin{array}{l}\[\text{log }\)x^2=y+3\\ \(\text{log }\)x^{}=y-1\(\end{array}\]\right\).


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