X(3 1/4,4 3/4,6 1/4) Y(10,9,5) is this linear or nonlinear

To determine whether the given set of points \( X(3 \frac{1}{4}, 4 \frac{3}{4}, 6 \frac{1}{4}) \) and \( Y(10, 9, 5) \) is linear or nonlinear, we can analyze the relationship between \( X \) and \( Y \). Specifically, we need to see if there's a linear relationship (i.e., \( Y = mX + b \)) between these variables.

1. **Convert the mixed fractions to improper fractions or decimals:**

   \[
   X = \left( 3 \frac{1}{4}, 4 \frac{3}{4}, 6 \frac{1}{4} \right) = \left( 3.25, 4.75, 6.25 \right)
   \]

2. **Check if the change in \( Y \) corresponds to a consistent change in \( X \).**
   
   We calculate the slope between each pair of points:
   
   \[
   \text{Slope between the first and second points} = \frac{9 - 10}{4.75 - 3.25} = \frac{-1}{1.5} = -\frac{2}{3}
   \]
   \[
   \text{Slope between the second and third points} = \frac{5 - 9}{6.25 - 4.75} = \frac{-4}{1.5} = -\frac{8}{3}
   \]

3. **Compare the slopes:**

   The slope between the first and second points is \(-\frac{2}{3}\), while the slope between the second and third points is \(-\frac{8}{3}\).

Since the slopes are not consistent, the points do not form a straight line, indicating that the relationship is **nonlinear**.

Would you like to see further details or have any questions?

Here are five related questions for further exploration:
1. What is the general method to determine linearity in a set of data points?
2. How can you use a graph to visually check for linearity?
3. What are some examples of nonlinear functions?
4. How do you find the equation of a line given two points?
5. How would you determine linearity if given more than three points?

**Tip:** When analyzing data for linearity, always compare slopes between consecutive points to ensure consistency.

Learn how to determine if a set of data points X(3 1/4, 4 3/4, 6 1/4) and Y(10, 9, 5) exhibit a linear relationship. Analyze slopes between points and understand the concept of linearity versus nonlinearity in mathematical relationships. Suitable for high school students studying algebra and linear equations.

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