Linear Modeling Word Problems

Overview

• Topic: Linear modeling — word problems using linear equations.
• Goal: Translate real-world situations to linear equations, find slope and predict values.

Key Concepts

• Independent variable (x): time (years) in examples.
• Dependent variable (y): value or sales (money) depending on time.
• Slope (m): rate of change = (y2 - y1) / (x2 - x1).
• Slope-intercept form: y = m x + b (b is y-intercept).
• Point-slope form: y - y1 = m (x - x1) when y-intercept is unknown.

Example 1: Depreciation of a Fax Machine

• Given:
  • Purchase price at 0 years: $2,700 -> point (0, 2700).
  • Value at 5 years: $100 -> point (5, 100).
• Steps:
  • Compute slope: m = (2700 - 100) / (0 - 5) = 2600 / (-5) = -520.
  • Use slope-intercept (y-intercept b = 2700): y = -520x + 2700.
  • Evaluate at x = 3: y = -520(3) + 2700 = -1560 + 2700 = 1140.
• Answer:
  • Value after 3 years = $1,140.

Example 2: Business Sales Growth

• Given:
  • Year 2 sales: $50,000 -> point (2, 50000).
  • Year 5 sales: $100,000 -> point (5, 100000).
• Steps:
  • Compute slope: m = (50000 - 100000) / (2 - 5) = (-50000) / (-3) = 50000/3.
  • Use point-slope form with point (2, 50000): y - 50000 = (50000/3)(x - 2).
  • Evaluate at x = 8: x - 2 = 6, so y - 50000 = (50000/3)6 = 500002 = 100000.
  • Solve: y = 100000 + 50000 = 150000.
• Answer:
  • Sales in year 8 = $150,000.

Table: Structured Example Summary

| Scenario | Points (x,y) | Slope (m) | Equation | Requested x | Result y | | Fax machine | (0, 2700), (5, 100) | -520 | y = -520x + 2700 | 3 | $1,140 | | Business sales | (2, 50000), (5, 100000) | 50000/3 | y - 50000 = (50000/3)(x-2) | 8 | $150,000 |

Key Terms and Definitions

• Slope: Change in y divided by change in x; indicates rate of change.
• Y-intercept (b): Value of y when x = 0.
• Point-slope form: Useful when slope known but y-intercept unknown.
• Dependent vs Independent: Dependent variable changes with independent variable.

Procedure / Steps To Solve Word Problems

• Identify independent (x) and dependent (y) variables.
• Convert given information into two points (x, y).
• Compute slope m = (y2 - y1) / (x2 - x1).
• Choose equation form:
  • If y-intercept known: use y = m x + b.
  • If not known: use point-slope form y - y1 = m (x - x1).
• Substitute requested x (or y) and solve.

Practice Problem (Given)

• Problem: Car bought for $188,000; after 5 years worth $10,500. When will it be worth $0?
• Hints:
  • Points: (0, 188000) and (5, 10500).
  • Find slope, write equation, set y = 0, solve for x.

Next Steps / Action Items

• Practice converting word data to points and equations.
• Solve the provided car problem using the outlined steps.
• Check units (years, dollars) when interpreting answers.