3-4.2 Application Problem P. 85 Answers

Tackling Application Problems in Chapter 3-4.2 (p. 85): A Comprehensive Guide

This article provides a detailed walkthrough of the application problems found on page 85 of Chapter 3-4.2, a section likely referring to a specific textbook or learning material. Because I do not have access to the specific textbook you are referring to, I will instead provide a general framework and examples for tackling application problems related to a common chapter topic at this point in a typical math or science curriculum: systems of linear equations and inequalities. This framework will adapt to other topics, provided you replace the specific examples with those from your textbook.

Remember to always refer to your textbook and class notes for the specific context and formulas needed to solve your particular problems. This article aims to provide a general methodology rather than specific solutions to your problems.

Understanding the Context: Systems of Linear Equations and Inequalities

Before diving into application problems, let's refresh our understanding of systems of linear equations and inequalities. These systems involve multiple equations or inequalities, each representing a relationship between variables. The goal is typically to find the values of the variables that satisfy all equations or inequalities simultaneously.

Types of Problems:

• Systems of Linear Equations: These involve finding the point(s) where two or more lines intersect. Solutions can be unique (one point), infinitely many (lines overlap), or no solution (lines are parallel). Methods for solving include substitution, elimination, and graphing.

• Systems of Linear Inequalities: These involve finding the region on a graph that satisfies all inequalities. The solution is often a shaded area representing the feasible region. Graphing is the primary method for solving.

Strategies for Solving Application Problems (Chapter 3-4.2, p. 85)

Successfully tackling application problems requires a systematic approach. Here's a step-by-step process:

1. Carefully Read and Understand the Problem

This step is often overlooked but is crucial. Identify:

• What is being asked? What are the unknowns you need to find?
• What information is given? Identify all relevant data and relationships.
• What are the constraints? Are there any limitations or restrictions on the variables?

2. Define Variables

Assign variables (x, y, z, etc.) to represent the unknowns you identified in step 1. Clearly define what each variable represents. For example:

• x = number of apples
• y = number of oranges

3. Translate the Problem into Equations or Inequalities

This is the most challenging step, translating the word problem into mathematical language. Look for keywords that indicate mathematical operations:

• "Sum," "total," "more than," "increased by" suggest addition (+)
• "Difference," "less than," "decreased by" suggest subtraction (-)
• "Product," "times," "of" suggest multiplication (×)
• "Quotient," "divided by" suggest division (÷)
• "Is," "equals," "is equal to" suggest equality (=)
• "Greater than," "at least" suggest greater than or equal to (≥)
• "Less than," "at most" suggest less than or equal to (≤)

4. Solve the System of Equations or Inequalities

Use appropriate methods to solve the system you've created. This might involve:

• Substitution: Solving one equation for one variable and substituting it into another.
• Elimination: Adding or subtracting equations to eliminate one variable.
• Graphing: Plotting the equations or inequalities on a graph to find the solution.

5. Check Your Solution

Always check your solution to ensure it makes sense in the context of the original problem. Does it satisfy all given conditions and constraints? Are the values realistic (e.g., can you have a negative number of apples)? If not, review your work to find any errors.

6. Write Your Answer Clearly

State your answer in a complete sentence, using the appropriate units. For example: "There are 15 apples and 10 oranges."

Example Application Problems and Solutions (Illustrative)

Let's illustrate this process with some examples, assuming the chapter involves systems of linear equations. Remember these are illustrative; replace them with your actual problems from page 85.

Example 1: Mixing Solutions

A chemist needs to mix a 10% acid solution with a 30% acid solution to obtain 100 liters of a 25% acid solution. How many liters of each solution should be used?

1. Understand the Problem: We need to find the number of liters of the 10% and 30% solutions.

2. Define Variables: Let x be the liters of 10% solution and y be the liters of 30% solution.

3. Translate to Equations:

   • Total volume: x + y = 100
   • Total acid: 0.10x + 0.30y = 0.25(100)

4. Solve: We can use elimination or substitution. Let's use elimination:

   • Multiply the first equation by -0.10: -0.10x - 0.10y = -10
   • Add this to the second equation: 0.20y = 15 => y = 75
   • Substitute y = 75 into x + y = 100: x + 75 = 100 => x = 25

5. Check: 25 + 75 = 100 (total volume). 0.10(25) + 0.30(75) = 2.5 + 22.5 = 25 (total acid). This checks out.

6. Answer: The chemist should use 25 liters of the 10% solution and 75 liters of the 30% solution.

Example 2: Ticket Sales

A school sold 300 tickets for a play. Adult tickets cost $5, and student tickets cost $3. If the total revenue was $1140, how many adult and student tickets were sold?

1. Understand: Find the number of adult and student tickets sold.

2. Define Variables: Let a be the number of adult tickets and s be the number of student tickets.

3. Translate:

   • Total tickets: a + s = 300
   • Total revenue: 5a + 3s = 1140

4. Solve: Use elimination or substitution. Let's use elimination:

   • Multiply the first equation by -3: -3a - 3s = -900
   • Add this to the second equation: 2a = 240 => a = 120
   • Substitute a = 120 into a + s = 300: 120 + s = 300 => s = 180

5. Check: 120 + 180 = 300 (total tickets). 5(120) + 3(180) = 600 + 540 = 1140 (total revenue).

6. Answer: The school sold 120 adult tickets and 180 student tickets.

Expanding the Scope: Advanced Application Problems

More advanced application problems might involve:

• Nonlinear equations: Equations that aren't straight lines (e.g., parabolas, circles). These often require more sophisticated solution techniques.
• Optimization problems: Finding the maximum or minimum value of a function subject to constraints. Linear programming techniques might be needed.
• Systems with three or more variables: These problems require more complex algebraic manipulation.

Strategies for Advanced Problems:

• Break the problem down: Divide complex problems into smaller, more manageable parts.
• Draw diagrams: Visual aids can help clarify relationships between variables.
• Use technology: Calculators or computer software can assist with solving complex systems.
• Consult resources: Refer to your textbook, class notes, or online resources for help.

Remember that persistence and practice are key to mastering application problems. The more problems you solve, the better you'll become at recognizing patterns, translating word problems into mathematical equations, and selecting appropriate solution methods. Don't be afraid to seek help from your teacher, classmates, or tutors if you're struggling. By systematically applying the strategies outlined above, you can confidently tackle even the most challenging application problems.