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Questions and Answers
What does the slope of a linear equation represent?
What does the slope of a linear equation represent?
The Y-intercept represents the total cost of an item or service.
The Y-intercept represents the total cost of an item or service.
False
What keyword indicates the unit rate in a word problem?
What keyword indicates the unit rate in a word problem?
per
The typical equation for a linear equation is y = _______ x + b.
The typical equation for a linear equation is y = _______ x + b.
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What is the first step in solving an equation?
What is the first step in solving an equation?
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Match the following parts of a linear equation with their meanings:
Match the following parts of a linear equation with their meanings:
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In the equation y = 8x + 12, what is the total cost if the boat is rented for 5 hours?
In the equation y = 8x + 12, what is the total cost if the boat is rented for 5 hours?
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Study Notes
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.
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Description
Learn about linear equations, slope, and Y-intercept in the context of unit rates and initial values. Understand how to interpret word problems and identify keywords.
