Unit Linear Relationships Homework 1 Answer Key

Unit Linear Relationships Homework 1 Answer Key: A Comprehensive Guide

This comprehensive guide provides answers and detailed explanations for a typical "Unit Linear Relationships Homework 1." It covers various aspects of linear relationships, including identifying them, graphing them, finding slopes and intercepts, writing equations, and solving real-world problems. Remember to always check your teacher's specific instructions and examples, as variations may exist.

Understanding Linear Relationships

Before diving into the answers, let's review the core concepts:

A linear relationship is a relationship between two variables where the change in one variable is directly proportional to the change in the other. This means that if you graph the relationship, it will form a straight line.

Key characteristics of linear relationships:

• Constant rate of change: The slope (steepness) of the line remains consistent throughout.
• Equation form: Linear relationships can be represented by the equation y = mx + b, where:
  • 'm' is the slope (rate of change).
  • 'b' is the y-intercept (the point where the line crosses the y-axis).
• Table representation: A table of values showing the relationship between x and y will reveal a constant difference in y for each consistent change in x.

Identifying Linear Relationships

A linear relationship can be identified through various methods:

• Graphical representation: If a graph of the data points forms a straight line, it indicates a linear relationship.
• Table of values: A consistent difference in the y-values for equal intervals in the x-values suggests linearity.
• Equation: An equation in the form y = mx + b is inherently a linear relationship.

Homework Problem Examples and Solutions

Let's explore common linear relationship problems found in Homework 1 and their solutions. These examples will cover a range of difficulty levels.

Problem 1: Identifying Linear Relationships from Graphs

(a) Graph A shows a straight line. Is this a linear relationship?

Answer: Yes. A straight line is the defining characteristic of a linear relationship.

(b) Graph B shows a curve. Is this a linear relationship?

Answer: No. A curve indicates a non-linear relationship. The rate of change is not constant.

(c) Graph C shows scattered points with no clear pattern. Is this a linear relationship?

Answer: No. Scattered points without a clear trend do not represent a linear relationship. There might be a correlation, but it's not linear.

Problem 2: Identifying Linear Relationships from Tables

Examine the following tables. Determine whether each represents a linear relationship.

(a) Table 1:

Answer: Yes. The difference in y-values (Δy) is consistently 2 for each increase of 1 in x-values (Δx). The slope (m = Δy/Δx) is 2.

(b) Table 2:

Answer: No. The difference in y-values is not consistent. This table represents an exponential relationship.

Problem 3: Finding the Slope and y-intercept

Find the slope (m) and y-intercept (b) for the following equation: y = 3x - 5

Answer:

• Slope (m): The slope is the coefficient of x, which is 3.
• y-intercept (b): The y-intercept is the constant term, which is -5. This means the line crosses the y-axis at the point (0, -5).

Problem 4: Writing the Equation of a Line

Write the equation of the line that passes through the points (2, 4) and (4, 10).

Solution:

1. Find the slope (m): m = (y₂ - y₁) / (x₂ - x₁) = (10 - 4) / (4 - 2) = 6 / 2 = 3

2. Use the point-slope form: y - y₁ = m(x - x₁) Using point (2, 4): y - 4 = 3(x - 2)

3. Simplify to slope-intercept form: y = 3x - 2

Problem 5: Solving Real-World Problems

A taxi charges a flat fee of $5 plus $2 per mile. Write an equation representing the total cost (y) based on the number of miles (x).

Answer:

y = 2x + 5

This equation represents a linear relationship where the slope (2) represents the cost per mile and the y-intercept (5) represents the flat fee.

Problem 6: Graphing Linear Equations

Graph the equation y = -2x + 4.

Solution:

1. Find the y-intercept: The y-intercept is 4. Plot the point (0, 4).

2. Find another point: Use the slope (-2). This means for every 1 unit increase in x, y decreases by 2 units. Starting from (0, 4), move 1 unit to the right and 2 units down, giving you the point (1, 2).

3. Draw a line: Draw a straight line through the points (0, 4) and (1, 2). Extend the line in both directions.

Problem 7: Interpreting the Slope and Intercept in Context

The equation C = 100 + 25n represents the total cost (C) of a phone plan where 'n' is the number of gigabytes used. Interpret the slope and y-intercept.

Answer:

• Slope (25): The slope represents the cost per gigabyte used. For each additional gigabyte, the cost increases by $25.

• Y-intercept (100): The y-intercept represents the fixed cost, regardless of the number of gigabytes used. This could be a base monthly fee.

Advanced Linear Relationship Concepts (Potentially in Homework 1)

Some homework assignments might include more advanced topics:

• Parallel and Perpendicular Lines: Understanding how slopes relate to parallel (same slope) and perpendicular (negative reciprocal slopes) lines.

• Systems of Linear Equations: Solving for the point of intersection between two lines, often using methods like substitution or elimination.

• Linear Inequalities: Graphing and solving inequalities involving linear expressions, represented by shaded regions on a graph.

Tips for Success with Linear Relationships

• Practice Regularly: The more you practice, the better you'll understand the concepts.
• Visualize: Graphs are powerful tools. Use them to visualize the relationships between variables.
• Understand the Context: Pay attention to the real-world applications and interpret the results within that context.
• Check your work: Always double-check your calculations and graphs to ensure accuracy.
• Seek help when needed: Don't hesitate to ask your teacher, classmates, or tutor for help if you're struggling.

This comprehensive guide provides a solid foundation for understanding and solving problems related to linear relationships. Remember to apply these concepts to your specific homework problems and consult your textbook or teacher for additional assistance. Good luck!