Linear Equations and Unit Rates

Linear Equations and Unit Rates

• A linear equation requires two things: slope (unit rate) and Y-intercept (initial value)
• Slope (unit rate) represents the cost of something per unit of time or item
• Y-intercept (initial value) is the additional cost before using the item or service

Interpreting Word Problems

• Identify keywords: "per" indicates unit rate (slope); "initial fee" or "extra cost" indicates Y-intercept (initial value)
• Associate money with Y and time or items with X

Setting Up Equations

• Typical equation: y = mx + b
• Slope (m) is the unit rate, and Y-intercept (b) is the initial value
• Plug in known values to set up the equation

Solving Equations

• Plug in known values to solve for unknowns
• Isolate the variable (X) by subtracting or dividing
• Remember to include units in the answer (e.g., hours, dollars)

Example Problem: Boat Rental

• Equation: y = 8x + 12 (slope = 8/hour,Y−intercept=8/hour, Y-intercept = 8/hour,Y−intercept=12 initial fee)
• If total cost is $120, how many hours was the boat rented?
  • Plug in 120 for y and solve for x: x = 13.5 hours
• If the boat is rented for 3 hours, what is the total cost?
  • Plug in 3 for x and solve for y: y = $36

Linear Equations and Unit Rates

• A linear equation consists of two essential components: slope (unit rate) and Y-intercept (initial value), which are used to describe a relationship between two variables.
• Slope (unit rate) represents a rate of change, specifying the cost of an item or service per unit of time.
• Y-intercept (initial value) is an additional cost or initial fee that occurs before using the item or service.

Interpreting Word Problems

• Identify keywords to determine the type of problem: "per" indicates a unit rate, while "initial fee" or "extra cost" indicates a Y-intercept.
• Associate variables correctly: pair monetary values with the y-axis and time or items with the x-axis.

Setting Up Equations

• The general form of a linear equation is y = mx + b, where m represents the slope (unit rate) and b represents the Y-intercept (initial value).
• Plug in known values to set up the equation, ensuring accurate representation of the problem.

Solving Equations

• Substitute known values into the equation to solve for unknowns.
• Isolate the variable (x) by performing operations such as subtraction, addition, multiplication, or division.
• Always include units in the final answer, such as hours, dollars, or miles.

Example Problem: Boat Rental

• The equation y = 8x + 12 represents a boat rental scenario, where the slope (unit rate) is 8perhourandtheY−intercept(initialvalue)isa8 per hour and the Y-intercept (initial value) is a 8perhourandtheY−intercept(initialvalue)isa12 initial fee.
• To find the number of hours the boat was rented, plug in the total cost ($120) and solve for x: x = 13.5 hours.
• To find the total cost for a 3-hour rental, plug in x = 3 and solve for y: y = $36.