Given The Table Of Values:$\[ \begin{tabular}{|c|c|c|c|c|} \hline $x$ & 2 & 4 & 6 & 8 \\ \hline $y$ & 197 & 140 & 150 & 126 \\ \hline \end{tabular} \\]Determine Which Equation Models The Data:A. \[$ Y = -11x + 205 \$\] B. \[$ Y =

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Introduction

In mathematics, linear equations are used to model real-world data and phenomena. Given a set of data points, we can determine which linear equation best models the data. In this article, we will explore how to model data with linear equations using a given table of values.

Understanding the Data

The table of values provided is as follows:

xx 2 4 6 8
yy 197 140 150 126

We are given four data points: (2, 197), (4, 140), (6, 150), and (8, 126). Our goal is to determine which linear equation models this data.

Linear Equations

A linear equation is an equation in which the highest power of the variable(s) is 1. In this case, we are looking for an equation of the form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

Calculating the Slope

To determine the slope of the linear equation, we can use the formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are two data points.

Let's calculate the slope using the first two data points: (2, 197) and (4, 140).

m=140−1974−2=−572=−28.5m = \frac{140 - 197}{4 - 2} = \frac{-57}{2} = -28.5

Calculating the Y-Intercept

Now that we have the slope, we can calculate the y-intercept using one of the data points. Let's use the first data point: (2, 197).

b=y−mx=197−(−28.5)(2)=197+57=254b = y - mx = 197 - (-28.5)(2) = 197 + 57 = 254

Equation of the Line

Now that we have the slope and y-intercept, we can write the equation of the line:

y=−28.5x+254y = -28.5x + 254

Comparing with the Given Options

We are given two options: A. y=−11x+205y = -11x + 205 and B. y=2x+100y = 2x + 100. Let's compare our equation with these options.

Option A: y=−11x+205y = -11x + 205

Our equation: y=−28.5x+254y = -28.5x + 254

The slopes are different, so this option is not a good fit.

Option B: y=2x+100y = 2x + 100

Our equation: y=−28.5x+254y = -28.5x + 254

The slopes are also different, so this option is not a good fit.

Conclusion

Based on our calculations, we can conclude that the equation y=−28.5x+254y = -28.5x + 254 models the data best. This equation has a slope of -28.5 and a y-intercept of 254.

Discussion

This case study demonstrates how to model data with linear equations using a given table of values. By calculating the slope and y-intercept, we can determine which linear equation best models the data. This is an important skill in mathematics and is used in many real-world applications.

References

  • [1] "Linear Equations" by Khan Academy
  • [2] "Modeling Data with Linear Equations" by Math Is Fun

Additional Resources

  • [1] "Linear Equations" by Wolfram MathWorld
  • [2] "Modeling Data with Linear Equations" by IXL Math

Introduction

In our previous article, we explored how to model data with linear equations using a given table of values. In this article, we will answer some frequently asked questions about modeling data with linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In this case, we are looking for an equation of the form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

Q: How do I calculate the slope of a linear equation?

A: To calculate the slope of a linear equation, you can use the formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are two data points.

Q: How do I calculate the y-intercept of a linear equation?

A: To calculate the y-intercept of a linear equation, you can use the formula:

b=y−mxb = y - mx

where (x,y)(x, y) is a data point and mm is the slope.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2.

Q: How do I determine which linear equation models a set of data?

A: To determine which linear equation models a set of data, you can calculate the slope and y-intercept of the equation using the data points. You can then compare the equation with the given options to determine which one is the best fit.

Q: What are some common mistakes to avoid when modeling data with linear equations?

A: Some common mistakes to avoid when modeling data with linear equations include:

  • Not using enough data points to accurately determine the slope and y-intercept
  • Not checking for outliers or anomalies in the data
  • Not considering the possibility of a non-linear relationship between the variables
  • Not using a consistent method for calculating the slope and y-intercept

Q: How do I use linear equations in real-world applications?

A: Linear equations are used in many real-world applications, including:

  • Modeling population growth and decline
  • Predicting stock prices and market trends
  • Analyzing the relationship between variables in a scientific experiment
  • Creating mathematical models of real-world phenomena

Conclusion

Modeling data with linear equations is an important skill in mathematics and is used in many real-world applications. By understanding how to calculate the slope and y-intercept of a linear equation, you can determine which equation models a set of data. Remember to avoid common mistakes and consider the possibility of a non-linear relationship between the variables.

Additional Resources

  • [1] "Linear Equations" by Khan Academy
  • [2] "Modeling Data with Linear Equations" by Math Is Fun
  • [3] "Linear Equations in Real-World Applications" by Wolfram MathWorld

Frequently Asked Questions

  • Q: What is the difference between a linear equation and a quadratic equation? A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2.
  • Q: How do I determine which linear equation models a set of data? A: To determine which linear equation models a set of data, you can calculate the slope and y-intercept of the equation using the data points. You can then compare the equation with the given options to determine which one is the best fit.
  • Q: What are some common mistakes to avoid when modeling data with linear equations? A: Some common mistakes to avoid when modeling data with linear equations include not using enough data points to accurately determine the slope and y-intercept, not checking for outliers or anomalies in the data, not considering the possibility of a non-linear relationship between the variables, and not using a consistent method for calculating the slope and y-intercept.