$\[ \begin{tabular}{ccccc} $x$ & 0 & 1 & 2 & 3 \\ \hline $y$ & 54 & 72 & 96 & 128 \\ \end{tabular} \\]Which Of The Following Equations Relates \[$ Y \$\] To \[$ X \$\] For The Values In The Table?Choose 1 Answer:A. \[$ Y =

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Linear Relationship Between x and y

Understanding the Table

The given table represents a set of values for x and y, where x ranges from 0 to 3, and y ranges from 54 to 128. To determine the relationship between x and y, we need to analyze the pattern in the table.

Analyzing the Pattern

Upon examining the table, we can observe that the values of y increase by a certain amount as the values of x increase by 1. Specifically:

  • When x increases from 0 to 1, y increases from 54 to 72, which is an increase of 18.
  • When x increases from 1 to 2, y increases from 72 to 96, which is an increase of 24.
  • When x increases from 2 to 3, y increases from 96 to 128, which is an increase of 32.

Identifying the Relationship

From the pattern observed in the table, we can see that the increase in y is not constant, but rather it increases by a certain amount each time x increases by 1. This suggests that the relationship between x and y is not a simple linear relationship, but rather a quadratic or exponential relationship.

Choosing the Correct Equation

Based on the pattern observed in the table, we can try to find an equation that relates y to x. Let's consider the following options:

A. y = 18x + 36 B. y = 24x + 24 C. y = 32x + 16 D. y = 36x + 18

Evaluating the Options

To determine which equation is correct, we can substitute the values of x and y from the table into each option and see which one satisfies the equation.

Option A: y = 18x + 36

  • When x = 0, y = 18(0) + 36 = 36, which is not equal to 54.
  • When x = 1, y = 18(1) + 36 = 54, which is equal to 54.
  • When x = 2, y = 18(2) + 36 = 72, which is equal to 72.
  • When x = 3, y = 18(3) + 36 = 90, which is not equal to 128.

Option B: y = 24x + 24

  • When x = 0, y = 24(0) + 24 = 24, which is not equal to 54.
  • When x = 1, y = 24(1) + 24 = 48, which is not equal to 72.
  • When x = 2, y = 24(2) + 24 = 72, which is equal to 72.
  • When x = 3, y = 24(3) + 24 = 96, which is equal to 96.

Option C: y = 32x + 16

  • When x = 0, y = 32(0) + 16 = 16, which is not equal to 54.
  • When x = 1, y = 32(1) + 16 = 48, which is not equal to 72.
  • When x = 2, y = 32(2) + 16 = 80, which is not equal to 96.
  • When x = 3, y = 32(3) + 16 = 112, which is not equal to 128.

Option D: y = 36x + 18

  • When x = 0, y = 36(0) + 18 = 18, which is not equal to 54.
  • When x = 1, y = 36(1) + 18 = 54, which is equal to 54.
  • When x = 2, y = 36(2) + 18 = 90, which is not equal to 96.
  • When x = 3, y = 36(3) + 18 = 126, which is not equal to 128.

Conclusion

Based on the analysis, we can see that option A, y = 18x + 36, is the only equation that satisfies the values of x and y in the table. Therefore, the correct equation that relates y to x is:

y = 18x + 36

This equation represents a linear relationship between x and y, where y increases by 18 for every 1 increase in x.
Q&A: Linear Relationship Between x and y

Frequently Asked Questions

In the previous article, we explored the linear relationship between x and y using the given table. Here are some frequently asked questions and their answers:

Q: What is the linear relationship between x and y?

A: The linear relationship between x and y is represented by the equation y = 18x + 36. This equation shows that y increases by 18 for every 1 increase in x.

Q: How do I determine the linear relationship between x and y?

A: To determine the linear relationship between x and y, you can analyze the pattern in the table. Look for the increase in y for every 1 increase in x. If the increase is constant, then the relationship is linear. If the increase is not constant, then the relationship may be quadratic or exponential.

Q: What is the significance of the constant term in the linear equation?

A: The constant term in the linear equation represents the y-intercept, which is the value of y when x is equal to 0. In the equation y = 18x + 36, the constant term is 36, which means that when x is equal to 0, y is equal to 36.

Q: Can I use the linear equation to predict the value of y for a given value of x?

A: Yes, you can use the linear equation to predict the value of y for a given value of x. Simply substitute the value of x into the equation and solve for y. For example, if you want to find the value of y when x is equal to 4, you can substitute x = 4 into the equation y = 18x + 36 and solve for y.

Q: What are some real-world applications of linear relationships?

A: Linear relationships have many real-world applications, including:

  • Modeling population growth
  • Predicting stock prices
  • Analyzing the relationship between variables in a scientific experiment
  • Creating a budget or financial plan

Q: Can I use the linear equation to solve a system of equations?

A: Yes, you can use the linear equation to solve a system of equations. If you have two linear equations with two variables, you can use substitution or elimination to solve for the variables. For example, if you have the equations y = 18x + 36 and y = 24x + 48, you can use substitution to solve for x and y.

Q: What are some common mistakes to avoid when working with linear relationships?

A: Some common mistakes to avoid when working with linear relationships include:

  • Assuming a linear relationship when it is not present
  • Failing to check for a constant rate of change
  • Not considering the y-intercept or slope of the line
  • Not using the correct equation or formula for the problem

Conclusion

In this Q&A article, we explored some frequently asked questions and their answers related to linear relationships. We discussed how to determine the linear relationship between x and y, the significance of the constant term in the linear equation, and some real-world applications of linear relationships. We also covered some common mistakes to avoid when working with linear relationships.