https://file.aiquickdraw.com/mathbot/file/1725999968428iv62aqf3.jpg

Let's go through each part of the question based on the image you've provided.

### Part (a):
The equation given is \( y = \frac{5}{2}x \).

We'll use this to fill in the table for the given values of \( x \):

\[
y = \frac{5}{2}x
\]

For \( x = 2 \):
\[
y = \frac{5}{2} \times 2 = 5
\]

For \( x = 3 \):
\[
y = \frac{5}{2} \times 3 = 7.5
\]

For \( x = 6 \):
\[
y = \frac{5}{2} \times 6 = 15
\]

So, the completed table for part (a) is:

| \( x \) | \( y \) |
|--------|--------|
| 2      | 5      |
| 3      | 7.5    |
| 6      | 15     |

**Is \( y \) proportional to \( x \)?**

Yes, \( y \) is proportional to \( x \), because the equation \( y = \frac{5}{2}x \) represents a direct proportionality between \( y \) and \( x \) (the relationship is linear, and there is no constant added to or subtracted from the expression). The constant of proportionality is \( \frac{5}{2} \), and as \( x \) increases, \( y \) increases proportionally.

### Part (b):
The equation given is \( y = 3.2x + 5 \).

We'll use this to fill in the table for the given values of \( x \):

For \( x = 1 \):
\[
y = 3.2 \times 1 + 5 = 3.2 + 5 = 8.2
\]

For \( x = 2 \):
\[
y = 3.2 \times 2 + 5 = 6.4 + 5 = 11.4
\]

For \( x = 4 \):
\[
y = 3.2 \times 4 + 5 = 12.8 + 5 = 17.8
\]

So, the completed table for part (b) is:

| \( x \) | \( y \)  |
|--------|----------|
| 1      | 8.2      |
| 2      | 11.4     |
| 4      | 17.8     |

**Is \( y \) proportional to \( x \)?**

No, \( y \) is **not** proportional to \( x \), because the equation \( y = 3.2x + 5 \) includes a constant term (+5). In a proportional relationship, the graph must pass through the origin (0,0), and there must be no additional constant added to or subtracted from the linear term.

---

Would you like more details or clarifications on any part of this?

### Follow-up Questions:
1. What is the general form of a direct proportionality equation?
2. How does adding a constant term affect the relationship between two variables?
3. Can you derive the constant of proportionality from a set of points in a linear relationship?
4. What happens to the proportionality if \( y = 3.2x \) had no constant added?
5. How can you verify if two variables are proportional from a table of values?

**Tip**: When checking for proportionality, always look for a consistent ratio between the two variables and ensure that the equation passes through the origin.

Is y proportional to x for the equations y = (5/2)x and y = 3.2x + 5? Fill in the tables and explain why or why not.

Solve problems determining if y is proportional to x for the equations y = (5/2)x and y = 3.2x + 5. Complete the tables and learn why y is or is not proportional to x. Suitable for Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation