The Equation of a Trend Line

Oct 13, 2025 by windailys.com 81 views

Introduction

In mathematics, solving equations is a crucial step in finding the equation of a trend line that passes through a set of points. Loren, in her work, attempted to solve the equation $10=\frac{19}{9}(149)+b$ for $b$ as part of her effort to find the equation of a trend line that passes through the points $(1,130)$ and $(10,149)$ . However, she made an error in her calculation. In this article, we will discuss the error Loren made and provide the correct solution to the equation.

The Equation of a Trend Line

The equation of a trend line is a mathematical representation of the relationship between two variables. It is typically represented in the form of $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept. The slope of the line represents the rate of change of the variable $y$ with respect to the variable $x$ . The y-intercept represents the point where the line intersects the y-axis.

Loren's Error

Loren's error occurred when she attempted to solve the equation $10=\frac{19}{9}(149)+b$ for $b$ . The correct equation to solve for $b$ is $10 = m(10) + b$ , where $m$ is the slope of the line. However, Loren used the point $(1,130)$ and $(10,149)$ to calculate the slope of the line, which resulted in the equation $10 = \frac{19}{9}(149) + b$ .

Calculating the Slope of the Line

To calculate the slope of the line, we use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points on the line. Using the points $(1,130)$ and $(10,149)$ , we can calculate the slope of the line as follows:

$m = \frac{149 - 130}{10 - 1}$ $m = \frac{19}{9}$

Solving for $b$

Now that we have the slope of the line, we can solve for $b$ using the equation $10 = m(10) + b$ . Substituting the value of $m$ into the equation, we get:

$10 = \frac{19}{9}(10) + b$ $10 = \frac{190}{9} + b$ $10 - \frac{190}{9} = b$ $\frac{90 - 190}{9} = b$ $-\frac{100}{9} = b$

Conclusion

In conclusion, Loren made an error in solving the equation $10=\frac{19}{9}(149)+b$ for $b$ . The correct equation to solve for $b$ is $10 = m(10) + b$ , where $m$ is the slope of the line. We calculated the slope of the line using the points $(1,130)$ and $(10,149)$ and found that $m = \frac{19}{9}$ . Solving for $b$ using the equation $10 = m(10) + b$ , we found that $b = -\frac{100}{9}$ .

The Correct Equation of the Trend Line

The correct equation of the trend line that passes through the points $(1,130)$ and $(10,149)$ is $y = \frac{19}{9}x - \frac{100}{9}$ .

The Importance of Checking Work

It is essential to check our work to ensure that we have made no errors. In this case, Loren's error occurred because she did not check her work carefully. By checking our work, we can avoid making mistakes and ensure that our solutions are correct.

Common Mistakes in Solving Equations

There are several common mistakes that people make when solving equations. Some of these mistakes include:

Not checking work: Failing to check our work can lead to errors in our solutions.
Not using the correct formula: Using the wrong formula can result in incorrect solutions.
Not substituting values correctly: Failing to substitute values correctly can lead to errors in our solutions.
Not simplifying expressions: Failing to simplify expressions can make it difficult to solve equations.

Conclusion

Q: What is the equation of a trend line?

A: The equation of a trend line is a mathematical representation of the relationship between two variables. It is typically represented in the form of $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept.

Q: How do I calculate the slope of the line?

A: To calculate the slope of the line, you use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points on the line.

Q: What is the y-intercept?

A: The y-intercept is the point where the line intersects the y-axis. It is represented by the value of $b$ in the equation $y = mx + b$ .

Q: How do I solve for $b$ ?

A: To solve for $b$ , you use the equation $y = mx + b$ and substitute the values of $m$ and $y$ into the equation. Then, you solve for $b$ .

Q: What is the correct equation of the trend line that passes through the points $(1,130)$ and $(10,149)$ ?

A: The correct equation of the trend line that passes through the points $(1,130)$ and $(10,149)$ is $y = \frac{19}{9}x - \frac{100}{9}$ .

Q: What is the value of $b$ in the equation $y = \frac{19}{9}x - \frac{100}{9}$ ?

A: The value of $b$ in the equation $y = \frac{19}{9}x - \frac{100}{9}$ is $-\frac{100}{9}$ .

Q: Why is it essential to check our work?

A: It is essential to check our work to ensure that we have made no errors. By checking our work, we can avoid making mistakes and ensure that our solutions are correct.

Q: What are some common mistakes people make when solving equations?

A: Some common mistakes people make when solving equations include:

Not checking work
Not using the correct formula
Not substituting values correctly
Not simplifying expressions

Q: How can I avoid making mistakes when solving equations?

A: To avoid making mistakes when solving equations, you should:

Check your work carefully
Use the correct formula
Substitute values correctly
Simplify expressions

Q: What is the importance of understanding the equation of a trend line?

A: Understanding the equation of a trend line is essential in various fields, including economics, finance, and science. It helps us to analyze and predict the behavior of variables and make informed decisions.

Q: How can I apply the concept of the equation of a trend line in real-life situations?

A: You can apply the concept of the equation of a trend line in real-life situations by:

Analyzing data to identify trends and patterns
Making predictions based on the equation of the trend line
Identifying areas for improvement and making informed decisions

Q: What are some real-life applications of the equation of a trend line?

A: Some real-life applications of the equation of a trend line include:

Predicting stock prices
Analyzing consumer behavior
Identifying trends in population growth
Making informed decisions in business and finance