Question

a. Proportional:Yes or no why?
b. What is the slope?
c. write the equation:

x y
3 2
6 4
9 6
12 8

Answer 1

To analyze the given set of points (x, y), let's evaluate them step by step.

### a. Proportional: Yes or No? Why?
**Yes,** the relationship is proportional. This is because the ratio of \(y\) to \(x\) is the same for all given points.

Calculating the ratios:
- For (3, 2): \(\frac{y}{x} = \frac{2}{3}\)
- For (6, 4): \(\frac{y}{x} = \frac{4}{6} = \frac{2}{3}\)
- For (9, 6): \(\frac{y}{x} = \frac{6}{9} = \frac{2}{3}\)
- For (12, 8): \(\frac{y}{x} = \frac{8}{12} = \frac{2}{3}\)

Since the ratio is the same (\(\frac{2}{3}\)) for all points, the relationship is indeed proportional.

### b. What is the slope?
The slope \(m\) can be calculated as the change in \(y\) divided by the change in \(x\).

Using two points, for example (3, 2) and (6, 4):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{6 - 3} = \frac{2}{3}
\]

So, the slope is \(\frac{2}{3}\).

### c. Write the equation:
The equation of a line in slope-intercept form is given by \(y = mx + b\). Since this relationship is proportional, it passes through the origin (0,0), meaning \(b = 0\).

Thus, using the slope we just calculated:
\[
y = \frac{2}{3}x
\]

### Summary
a. Yes, it is proportional (constant ratio \(\frac{2}{3}\)).
b. The slope is \(\frac{2}{3}\).
c. The equation is \(y = \frac{2}{3}x\).