# 55 Unit 2 Linear Functions Homework 5 Answer Key

## Unit 2 Linear Functions Homework 5 Answer Key

### Introduction

Linear functions are an essential part of mathematics, and understanding them is crucial for success in various fields such as engineering, finance, and computer science. In unit 2 of your curriculum, you have been introduced to the concept of linear functions and have explored different aspects of them. Homework assignments are an integral part of the learning process, allowing you to practice what you have learned and solidify your understanding. In this article, we will provide you with the answer key to unit 2 linear functions homework 5.

### Question 1: Finding the Slope

To find the slope of a linear function, you need to determine the change in the y-values divided by the change in the corresponding x-values. In this question, you were given two points, (3, 7) and (5, 11), and asked to find the slope of the line passing through them. By applying the slope formula, we can calculate the slope as follows:

Slope = (11 - 7) / (5 - 3) = 4 / 2 = **2**

### Question 2: Graphing a Linear Function

Graphing linear functions allows us to visualize the relationship between the input and output values. In this question, you were given a linear function in slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept. The equation provided was y = 2x + 3. To graph this function, follow these steps:

- Plot the y-intercept, which is the point (0, 3).
- Use the slope to find additional points. Since the slope is 2, you can move up 2 units and right 1 unit from the y-intercept to find another point, which is (1, 5).
- Connect the two points with a straight line.

Your graph should resemble a line that passes through the points (0, 3) and (1, 5).

### Question 3: Writing the Equation of a Line

A linear function can be represented by an equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. In this question, you were given a line with a slope of 3 passing through the point (2, 5). To write the equation of this line, use the given slope and point as follows:

y = mx + b

5 = 3(2) + b

5 = 6 + b

b = 5 - 6

b = -1

The equation of the line is y = 3x - 1.

### Question 4: Solving a System of Linear Equations

A system of linear equations consists of two or more equations that are solved simultaneously. In this question, you were given the following system of equations:

2x + y = 7

3x - y = 4

To solve this system, you can use the substitution or elimination method. Let's solve it using the elimination method:

- Add the two equations together to eliminate the y variable:
- Divide both sides of the equation by 5:
- Substitute the value of x into one of the original equations to solve for y:

(2x + y) + (3x - y) = 7 + 4

5x = 11

x = 11/5

2(11/5) + y = 7

22/5 + y = 7

y = 7 - 22/5

y = 35/5 - 22/5

y = 13/5

The solution to the system of equations is x = 11/5 and y = 13/5.

### Question 5: Finding the x-Intercept and y-Intercept

The x-intercept and y-intercept are the points where a line crosses the x-axis and y-axis, respectively. In this question, you were given a linear function in standard form, ax + by = c, and asked to find the x-intercept and y-intercept. The equation provided was 3x - 2y = 6. To find the x-intercept, set y = 0 and solve for x:

3x - 2(0) = 6

3x = 6

x = 2

The x-intercept is (2, 0).

To find the y-intercept, set x = 0 and solve for y:

3(0) - 2y = 6

-2y = 6

y = -3

The y-intercept is (0, -3).

### Question 6: Finding the Domain and Range

The domain of a function refers to the set of all possible input values, while the range refers to the set of all possible output values. In this question, you were given a linear function and asked to find its domain and range. The function provided was y = -2x + 4. To find the domain, consider the restrictions on the input variable, which is x in this case. Since there are no specific restrictions mentioned, the domain is all real numbers (-∞, +∞). To find the range, observe the behavior of the output variable, which is y. The coefficient of x is -2, indicating that the function is decreasing. Therefore, the range is also all real numbers (-∞, +∞).

### Question 7: Determining Parallel and Perpendicular Lines

Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. In this question, you were given a linear function and asked to determine whether two lines are parallel or perpendicular. The function provided was y = 2x + 3 for the first line and y = -1/2x + 4 for the second line. By comparing the slopes, we can determine the relationship:

Slope of the first line: 2

Slope of the second line: -1/2

Since the slopes are not equal and their product is not -1, the lines are neither parallel nor perpendicular.

### Question 8: Solving Word Problems

Linear functions can be applied to solve various real-world problems. In this question, you were presented with a word problem and asked to solve it using a linear function. The problem might involve finding the cost, distance, time, or any other variable related to the situation at hand. To solve word problems, it is important to identify the relevant quantities, establish the relationships between them, and formulate an equation based on the given information. Then, solve the equation to find the desired solution.

### Question 9: Interpreting Slope and Intercept

The slope and intercept of a linear function provide valuable insights into the relationship between the input and output values. In this question, you were given a linear function and asked to interpret the meaning of its slope and intercept in the context of a real-world scenario. The interpretation might involve understanding the rate of change, initial value, direction, or any other characteristic related to the situation being described. To interpret the slope and intercept, consider their numerical values and how they relate to the problem at hand.

### Question 10: Applying Linear Functions in Different Fields

Linear functions have numerous applications in various fields, including engineering, finance, and computer science. In this question, you were presented with scenarios from different domains and asked to apply the concepts of linear functions to solve the problems. By understanding how linear functions can be utilized in different contexts, you will develop a deeper appreciation for their practical significance.

### Conclusion

Unit 2 of your curriculum has provided you with a solid foundation in linear functions. By completing homework 5 and reviewing the answer key, you have reinforced your understanding of key concepts and enhanced your problem-solving skills. Remember to practice regularly and seek clarification whenever needed to further strengthen your grasp on linear functions. With this knowledge, you will be well-equipped to tackle more complex mathematical concepts and excel in various academic and professional endeavors.