Determining The Value Of A For A Linear Function

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In mathematics, understanding linear functions is crucial, especially when dealing with rates of change. This article delves into a specific problem where we need to find the value of a variable to ensure that a given data set represents a linear function with a particular rate of change. Let's explore the problem, the concepts involved, and the step-by-step solution.

The Problem

We are given a table of data that relates two variables, x and y. The table has the following entries:

x y
3 13
4 a
5 23

The task is to find the value of a such that the data in the table represents a linear function with a rate of change of +5. This means that for every increase of 1 in x, the value of y increases by 5. Understanding how to solve this problem involves grasping the fundamental concepts of linear functions and rates of change.

Understanding Linear Functions

In the realm of mathematics, a linear function is defined as a function that forms a straight line when graphed on a coordinate plane. Linear functions are characterized by a constant rate of change, meaning that the change in the dependent variable (y) is proportional to the change in the independent variable (x). This constant rate of change is also known as the slope of the line. The most common form of a linear function is the slope-intercept form, represented as:

y = mx + b

where:

  • y is the dependent variable
  • x is the independent variable
  • m is the slope or rate of change
  • b is the y-intercept (the point where the line crosses the y-axis)

The Rate of Change (Slope)

The rate of change, often referred to as the slope, is a crucial characteristic of a linear function. It quantifies how much the dependent variable (y) changes for each unit change in the independent variable (x). In other words, it measures the steepness and direction of the line. The rate of change (m) can be calculated using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

where (x₁, y₁) and (x₂, y₂) are two distinct points on the line. This formula calculates the change in y divided by the change in x, giving us the constant rate at which the line rises or falls.

Properties of Linear Functions

Linear functions have several key properties that make them essential in various mathematical and real-world applications. One of the most significant properties is the constant rate of change, which means that the slope remains the same throughout the entire line. This uniformity simplifies analysis and prediction.

Another important property is that linear functions can be uniquely defined by any two points on the line. Knowing just two points, we can determine the slope and the y-intercept, thereby defining the entire function. This characteristic is invaluable in modeling situations where two data points are known, and the relationship between the variables is linear.

Real-World Applications

Linear functions are widely used in various fields due to their simplicity and predictability. They are commonly used to model scenarios where there is a constant relationship between two variables. For example, the relationship between the number of hours worked and the amount earned at a fixed hourly wage is linear. Similarly, the distance traveled at a constant speed over time can be modeled using a linear function.

In economics, linear functions are used to represent cost and revenue models. For instance, a company's total cost can be modeled as a linear function of the number of units produced, where the slope represents the variable cost per unit, and the y-intercept represents the fixed costs. Understanding these applications highlights the practical importance of linear functions in everyday life and various professional domains.

Applying the Rate of Change

In our problem, we are given that the rate of change is +5. This means that for every unit increase in x, the value of y increases by 5. We can use this information to find the value of a. Let's analyze the given data points.

We have two complete points: (3, 13) and (5, 23). We can use these points to verify the rate of change. The change in x is 5 - 3 = 2, and the change in y is 23 - 13 = 10. The rate of change, therefore, is 10 / 2 = 5, which confirms the given rate of change.

Now, we need to find the value of a in the point (4, a). We can use either of the given points (3, 13) or (5, 23) to find a. Let's use the point (3, 13) and the fact that the rate of change is 5.

From x = 3 to x = 4, there is an increase of 1. Therefore, the value of y should increase by 5. So, we add 5 to the y-value of the point (3, 13):

a = 13 + 5
a = 18

Alternatively, we can use the point (5, 23) and work backward. From x = 5 to x = 4, there is a decrease of 1. Therefore, the value of y should decrease by 5. So, we subtract 5 from the y-value of the point (5, 23):

a = 23 - 5
a = 18

Both methods give us the same value for a, which is 18. This confirms that the value of a must be 18 for the data to represent a linear function with a rate of change of +5.

Step-by-Step Solution

To recap, here’s a step-by-step solution to the problem:

  1. Identify the Given Information:
    • We have a table with points (3, 13), (4, a), and (5, 23).
    • The rate of change is +5.
  2. Verify the Rate of Change (Optional):
    • Use the points (3, 13) and (5, 23) to verify the rate of change: (23 - 13) / (5 - 3) = 10 / 2 = 5.
  3. Use the Rate of Change to Find a:
    • From x = 3 to x = 4, there is an increase of 1 in x, so y should increase by 5.
    • a = 13 + 5 = 18
    • Alternatively, from x = 5 to x = 4, there is a decrease of 1 in x, so y should decrease by 5.
    • a = 23 - 5 = 18
  4. State the Solution:
    • The value of a must be 18.

Conclusion

In conclusion, to ensure that the given data represents a linear function with a rate of change of +5, the value of a must be 18. This problem demonstrates the importance of understanding linear functions, the rate of change, and how to apply these concepts to solve mathematical problems. By carefully analyzing the data and using the properties of linear functions, we can determine unknown values and ensure that the function behaves as expected. Understanding these principles is crucial for various applications in mathematics and real-world scenarios. By solving this problem, we reinforce our understanding of linear functions and their applications, ensuring that we can tackle similar challenges with confidence.

Therefore, the correct answer is C. a = 18.

Discussion Category: Mathematics

This problem falls under the category of mathematics, specifically linear functions and algebra. It involves understanding the concept of the rate of change (slope) and applying it to find an unknown value in a data set. The problem requires a solid grasp of linear equations and their properties, making it a fundamental topic in algebraic studies. Problems like this are common in introductory algebra courses and are crucial for building a strong foundation in mathematical reasoning and problem-solving. Understanding linear functions is not only essential for academic success but also for practical applications in various fields, such as science, economics, and engineering.