Intermediate Algebra
Unit 2 Overview: Solving Linear Equations
2.1 Use a General Strategy to Solve Linear Equations
2.2 Use a Problem Solving Strategy
2.3 Solve a Formula for a Specific Variable
2.4 Solve Mixture and Uniform Motion Applications
2.5 Solve Linear Inequalities
2.6 Solve Compound Inequalities
2.7 Solve Absolute Value Inequalities

📘intermediate algebra review

2.2 Use a Problem Solving Strategy

Last Updated on June 24, 2024

Problem-solving in algebra is all about breaking down complex issues into manageable steps. By understanding the given info, assigning variables, and translating problems into equations, you can tackle even the toughest questions. This systematic approach helps you solve and interpret solutions effectively.

Percentages and financial applications are crucial real-world uses of algebra. From calculating discounts and markups to computing simple interest, these skills are essential for personal finance. Understanding how to work with percentages and apply them to various scenarios will serve you well in everyday life.

Problem Solving Strategy

Systematic problem-solving approach

  • Understand the given information
    • Identify known quantities (prices, ages, distances)
    • Determine the unknown quantity to solve for (total cost, future age, arrival time)
  • Assign variables to unknown quantities
    • Choose meaningful variable names (x for the unknown value, t for time)
    • Define variables clearly to avoid confusion
  • Translate the problem into mathematical equations
    • Express relationships between known and unknown quantities using mathematical operations (addition, subtraction, multiplication, division)
    • Set up equations based on the problem statement (if John is 5 years older than Amy, then John's age = Amy's age + 5)
  • Solve the equations using appropriate algebraic techniques
    • Isolate the variable of interest by performing inverse operations
    • Apply inverse operations to solve for the unknown quantity (addition and subtraction, multiplication and division)
  • Interpret the solution in the context of the original problem
    • Check if the solution makes sense (a negative age or distance is not realistic)
    • Provide a clear answer to the question asked in the appropriate units (years, dollars, miles)
  • Use estimation to verify the reasonableness of the solution

Algebraic techniques for word problems

  • Identify the type of word problem
    • Recognize patterns and common problem types (age problems, distance-rate-time problems, mixture problems)
  • Extract relevant information from the problem statement
    • Identify given numbers and their relationships (John's age is 3 times Amy's age)
    • Determine the quantity to be found (the sum of their ages in 5 years)
  • Represent the problem using algebraic expressions or equations
    • Assign variables to unknown quantities (let x represent Amy's current age)
    • Express relationships between quantities using mathematical symbols (John's age = 3x)
  • Apply appropriate algebraic techniques to solve the equations
    • Simplify expressions by combining like terms
    • Isolate the variable of interest by performing inverse operations
    • Use inverse operations to solve for the unknown quantity (addition and subtraction, multiplication and division)
  • Check the solution and interpret it in the context of the problem
    • Verify that the solution satisfies the given conditions (substitute the value back into the original equations)
    • Provide a clear and concise answer to the question asked in the appropriate units (Amy is currently 10 years old)
  • Use visualization techniques (e.g., diagrams or graphs) to better understand complex problems

Enhanced problem-solving strategies

  • Apply logical reasoning to break down complex problems into smaller, manageable steps
  • Use a step-by-step approach to systematically work through each part of the problem
  • Employ critical thinking skills to analyze the problem from different angles and consider alternative solutions

Percentages and Financial Applications

Percentages in real-world scenarios

  • Understand the concept of percentages
    • Recognize that a percentage represents a fraction out of 100 (25% is equivalent to 25/100 or 0.25)
    • Convert between percentages, decimals, and fractions (75% = 0.75 = 3/4)
  • Calculate discounts
    • Determine the discount amount using the formula: Discount=Original Price×Discount Percentage\text{Discount} = \text{Original Price} \times \text{Discount Percentage} (a 50itemwitha2050 item with a 20% discount has a discount amount of 10)
    • Find the discounted price by subtracting the discount from the original price (the discounted price is $40)
  • Compute markups
    • Calculate the markup amount using the formula: Markup=Cost×Markup Percentage\text{Markup} = \text{Cost} \times \text{Markup Percentage} (a product with a cost of 20andamarkupof5020 and a markup of 50% has a markup amount of 10)
    • Determine the selling price by adding the markup to the cost (the selling price is $30)
  • Apply tax calculations
    • Compute the tax amount using the formula: Tax=Pre-tax Amount×Tax Rate\text{Tax} = \text{Pre-tax Amount} \times \text{Tax Rate} (a 100itemwithan8100 item with an 8% tax rate has a tax amount of 8)
    • Find the total cost by adding the tax to the pre-tax amount (the total cost is $108)

Simple interest calculations

  • Understand the components of simple interest
    • Principal (PP): The initial amount borrowed or invested (1000loan,1000 loan, 500 investment)
    • Interest Rate (rr): The percentage of the principal charged as interest, usually expressed as an annual rate (5% per year, 3.5% annually)
    • Time (tt): The duration of the loan or investment, typically measured in years (2 years, 18 months)
  • Use the simple interest formula: I=P×r×tI = P \times r \times t
    • II represents the interest earned or paid
    • Multiply the principal by the interest rate and time to calculate the interest (1000×0.05×2=1000 \times 0.05 \times 2 = 100)
  • Solve for the total amount (AA) using the formula: A=P+IA = P + I
    • Add the principal and interest to find the total amount after the specified time (1000+1000 + 100 = $1100)
  • Apply the simple interest formula to various financial scenarios
    • Calculating interest earned on investments (a 2000investmentat42000 investment at 4% annual interest for 3 years earns 240 in interest)
    • Determining interest paid on loans (a 5000loanat65000 loan at 6% annual interest for 2 years requires 600 in interest payments)

Key Terms to Review (22)

Like Terms: Like terms are algebraic expressions that have the same variable or combination of variables raised to the same power. They can be combined by adding or subtracting their coefficients, as they represent the same quantity in an expression.
Algebraic Expressions: Algebraic expressions are mathematical representations that combine variables, numbers, and operations to represent quantitative relationships. They are the fundamental building blocks used in algebra to model and solve a wide range of problems.
Variable: A variable is a symbol or letter that represents an unknown or changeable value in a mathematical expression, equation, or function. Variables are used to generalize and represent a range of possible values, allowing for the exploration of relationships and the solution of problems.
Markups: Markups refer to the amount added to the cost of a product or service to determine the selling price. It is a crucial concept in pricing strategies and profitability analysis, as it allows businesses to generate revenue and cover their expenses while maintaining a competitive edge in the market.
Problem Solving Strategy: A problem solving strategy is a systematic approach used to identify and address a problem or challenge. It involves a series of steps or techniques that guide an individual or a group towards finding a solution or resolution to the problem at hand.
Tax Calculations: Tax calculations refer to the process of determining the amount of tax an individual or entity owes to the government based on their income, purchases, or other taxable activities. This term is particularly relevant in the context of problem-solving strategies, as understanding tax calculations is essential for making informed financial decisions and accurately managing one's tax obligations.
Simple Interest: Simple interest is a method of calculating the amount of interest earned on a principal amount over a given period of time. It is a straightforward calculation based on the principal, interest rate, and time period.
Step-by-Step Approach: A step-by-step approach is a methodical problem-solving strategy that involves breaking down a complex problem into smaller, manageable steps. This systematic approach helps individuals understand the problem, identify the necessary information, and work through the solution in a logical and organized manner.
Estimation: Estimation is the process of approximating or determining a value, quantity, or outcome without the need for precise measurement or calculation. It involves making an informed judgment or educated guess based on available information and past experiences.
Logical Reasoning: Logical reasoning is the process of using rational, step-by-step thinking to arrive at a conclusion or make a decision. It involves analyzing information, identifying patterns, and drawing logical inferences to solve problems or reach sound judgments.
Principal: The principal is the original amount of money that is borrowed or invested, excluding any interest or additional charges. It serves as the foundation upon which interest and other financial calculations are based.
Percentages: Percentages are a way to express a part of a whole as a fraction of 100. They are commonly used to represent proportions, rates, and comparisons in various contexts, including problem-solving strategies.
Inverse Operations: Inverse operations are pairs of mathematical operations that undo each other, effectively reversing the original operation. They are essential in solving equations and formulas by isolating the variable of interest.
Algebraic Techniques: Algebraic techniques refer to the set of mathematical operations and methods used to manipulate and solve algebraic expressions, equations, and problems. These techniques are essential for understanding and applying various problem-solving strategies in mathematics.
Critical Thinking: Critical thinking is the objective analysis and evaluation of an issue in order to form a judgment. It involves the ability to think clearly and rationally, consider multiple perspectives, and make well-reasoned decisions.
Interest Rate: The interest rate is the amount charged, expressed as a percentage of the principal, by a lender to a borrower for the use of assets. It is a key factor in determining the cost of borrowing money and the return on saving or investing.
Discounts: Discounts refer to the reduction in the original price of a product or service, offered to customers as an incentive or promotion. They are a common pricing strategy used by businesses to increase sales, attract new customers, and retain existing ones.
Word Problems: Word problems are mathematical problems presented in the form of a written description, often involving real-world scenarios. They require the solver to extract the relevant information, identify the appropriate mathematical operations, and then apply them to find the solution.
Systematic Problem-Solving Approach: A systematic problem-solving approach is a structured and organized method for addressing and solving complex problems. It involves breaking down a problem into manageable steps, analyzing the underlying causes, and implementing a well-planned solution to achieve the desired outcome effectively.
Unknown Quantity: An unknown quantity refers to a variable or an element in a mathematical expression or equation whose value is not initially known, but can be determined through problem-solving strategies and techniques.
Mathematical Equations: Mathematical equations are symbolic representations of relationships between quantities, variables, and operations. They are fundamental tools used in problem-solving and quantitative analysis across various disciplines, including algebra, physics, and engineering.
Visualization: Visualization is the process of creating mental images or diagrams to better understand and solve problems. It helps in breaking down complex information into simpler, more manageable parts, allowing individuals to see relationships and patterns that may not be immediately apparent through raw data alone.