Modeling with Linear Equations: Fresh Take

  • Create and use linear equations and formulas to solve practical problems involving unknown quantities, dimensions, and distances.

Setting up a Linear Equation to Solve a Real-World Application

The Main Idea

  • Modeling Real-World Situations:
    • Identify known and unknown quantities
    • Assign variables to represent unknowns
    • Translate verbal descriptions into mathematical expressions
  • Components of Linear Equations:
    • Slope ([latex]m[/latex]): represents rate of change or variable cost
    • [latex]y[/latex]-intercept ([latex]b[/latex]): represents initial value or fixed cost
    • General form: [latex]y = mx + b[/latex]
  • Types of Real-World Applications:
    • Cost analysis (fixed vs. variable costs)
    • Revenue and profit modeling
    • Comparison of different options or plans
    • Inventory management
    • Time and distance problems

Key Steps in Problem-Solving

  1. Identify known and unknown quantities
  2. Assign variables to unknowns
  3. Express relationships between variables
  4. Write the equation
  5. Solve the equation
  6. Interpret the solution in context

 

There are two cell phone companies that offer different packages. Company A charges a monthly service fee of [latex]$34[/latex] plus [latex]$0.04[/latex]/min talk-time. Company B charges a monthly service fee of [latex]$45[/latex] plus [latex]$0.03[/latex]/min talk-time. Use [latex]x[/latex] for your variable.a) Write an equation that models the monthly cost for company A.b) Write an equation that models the monthly cost for company B.C) If the average number of minutes used each month is [latex]1,162[/latex], how much is the monthly cost for each company?

A cash register contains only five dollar and ten dollar bills. It contains twice as many five’s as ten’s and the total amount of money in the cash register is [latex]560[/latex] dollars. How many ten’s are in the cash register?

The following video shows another example of using linear equations to model and compare two cell phone plans.

You can view the transcript for “Write Linear Equations to Model and Compare Cell Phone Plans with Data Usage” here (opens in new window).

Watch the following video for another example of using a linear equation in one variable to solve an application.

You can view the transcript for “Solve a Coin Problem Using an Equation in One Variable” here (opens in new window).

Using a Formula to Solve a Real-World Application

The Main Idea

  • Application of Known Formulas:
    • Identify the appropriate formula for the given problem
    • Substitute known values into the formula
    • Solve for unknown variables
  • Common Formulas:
    • Perimeter of a rectangle: [latex]P = 2L + 2W[/latex]
    • Area of a rectangle: [latex]A = LW[/latex]
    • Volume of a rectangular solid: [latex]V = LWH[/latex]
    • Distance-Rate-Time: [latex]d = rt[/latex]
  • Multi-Step Problem Solving:
    • Break down complex problems into manageable steps
    • Use multiple formulas or equations when necessary
    • Express one unknown in terms of another when needed

Key Problem-Solving Steps

  1. Identify the given information and the unknown(s)
  2. Select the appropriate formula(s)
  3. Substitute known values into the formula
  4. Solve the equation for the unknown(s)
  5. Check the solution and interpret in context
  6. Consider unit conversions if necessary

Strategies for Solving Formula-Based Problems

  1. Visualization:
    • Draw diagrams or sketches to represent the problem
    • Label known and unknown quantities
  2. Unit Consistency:
    • Ensure all units are consistent before calculation
    • Convert units if needed (e.g., minutes to hours)
  3. Algebraic Manipulation:
    • Rearrange formulas to isolate the unknown variable
    • Use properties of equality to solve equations
  4. Multiple Equations:
    • Set up multiple equations for problems with more than one unknown
    • Use substitution or elimination methods to solve systems
  5. Reasonableness Check:
    • Verify if the solution makes sense in the context of the problem
    • Use estimation to check if the result is in the right ballpark

Common Pitfalls to Avoid

  1. Misidentifying the appropriate formula
  2. Forgetting to convert units
  3. Neglecting to consider all given information
  4. Misinterpreting the meaning of variables in context
  5. Not checking the solution for reasonableness
On Saturday morning, it took Jennifer [latex]3.6[/latex] hours to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer [latex]4[/latex] hours to return home. Her speed was [latex]5[/latex] mi/h slower on Sunday than on Saturday. What was her speed on Sunday?

Find the dimensions of a rectangle given that the perimeter is [latex]110[/latex] cm. and the length is [latex]1[/latex] cm. more than twice the width.

A game room has a perimeter of [latex]70[/latex] ft. The length is five more than twice the width. How many ft2 of new carpeting should be ordered?