Question

the table shows the linear relationship between the variables x and y
5 16.5
7 21.5
11 31.5
what are the slope y intercept forms of the linear relationships

A slope 2, y intercept (0, 6.5)
B slope -2, y intercept (4,0)
C slope -2.5, y intercept (4,0)
D slope 2.5, y intercept (0,4)

Answer 1

To determine the slope-intercept form of the linear relationship between the variables \( x \) and \( y \), we need to find the slope \( m \) and the y-intercept \( b \) of the line represented by the points given in the table.

The points are:
- \( (5, 16.5) \)
- \( (7, 21.5) \)
- \( (11, 31.5) \)

**Step 1: Calculate the slope \( m \)**

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \( (5, 16.5) \) and \( (7, 21.5) \):

\[
m = \frac{21.5 - 16.5}{7 - 5} = \frac{5}{2} = 2.5
\]

**Step 2: Use the slope to find the y-intercept \( b \)**

We can use the slope and one of the points to solve for \( b \) in the equation \( y = mx + b \).

Using the point \( (5, 16.5) \):

\[
16.5 = 2.5(5) + b
\]
\[
16.5 = 12.5 + b
\]
\[
b = 16.5 - 12.5 = 4
\]

So, the y-intercept is \( (0, 4) \).

**Final Equation:**

Thus, the equation in slope-intercept form is:

\[
y = 2.5x + 4
\]

**Conclusion:**

The slope is \( 2.5 \) and the y-intercept is \( (0, 4) \).

So the correct answer is: **D slope 2.5, y intercept (0, 4)**.