homework lesson 11 equations for proportional relationships answer key

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12-01-2025

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This answer key provides solutions and explanations for common problems encountered in Lesson 11 on equations for proportional relationships. Remember to always show your work and understand the underlying concepts. This key is intended to help you check your understanding, not to replace your own problem-solving efforts.

Understanding Proportional Relationships

Before diving into the answers, let's review the core concept: a proportional relationship exists between two variables when their ratio remains constant. This constant ratio is often represented by the constant of proportionality, often denoted as 'k'. The equation representing a proportional relationship is typically written as y = kx, where 'y' and 'x' are the variables and 'k' is the constant of proportionality.

Sample Problems and Solutions

This section will cover several example problems, demonstrating different approaches to solving for unknowns in proportional relationships. Remember to substitute your specific problem values for the examples provided.

Problem Type 1: Finding the Constant of Proportionality (k)

Problem: If y varies proportionally with x, and y = 12 when x = 4, find the constant of proportionality.

Solution:

1. Write the equation: y = kx
2. Substitute the given values: 12 = k * 4
3. Solve for k: k = 12 / 4 = 3

Therefore, the constant of proportionality is 3.

Problem Type 2: Writing the Equation

Problem: A car travels at a constant speed. It travels 150 miles in 3 hours. Write an equation to represent the distance (d) traveled in terms of time (t).

Solution:

1. Find the constant of proportionality (speed): Speed = Distance/Time = 150 miles / 3 hours = 50 miles/hour. This is our 'k' value.
2. Write the equation: d = 50t (where d is distance in miles and t is time in hours)

Therefore, the equation representing the relationship is d = 50t.

Problem Type 3: Solving for an Unknown Variable

Problem: Using the equation from the previous problem (d = 50t), how far will the car travel in 5 hours?

Solution:

1. Substitute the known value: d = 50 * 5
2. Solve for the unknown: d = 250 miles

Therefore, the car will travel 250 miles in 5 hours.

Problem Type 4: Interpreting Graphs of Proportional Relationships

Graphs of proportional relationships always pass through the origin (0,0) and show a straight line. The slope of the line represents the constant of proportionality (k).

Problem: A graph shows a proportional relationship between the number of apples (x) and their total cost (y). The line passes through the point (2, 6). What is the cost of 5 apples?

Solution:

1. Find k: k = y/x = 6/2 = 3 (The cost per apple is $3)
2. Write the equation: y = 3x
3. Solve for y when x = 5: y = 3 * 5 = 15

Therefore, the cost of 5 apples is $15.

Further Practice

To solidify your understanding, practice more problems from your textbook or worksheet. Focus on understanding the relationship between the variables and the constant of proportionality. If you continue to struggle with specific problem types, review the relevant sections of your textbook or seek assistance from your teacher or a tutor. Remember, consistent practice is key to mastering this concept.