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To find \( r \), the \( y \)-intercept of the line that passes through the points \((-3, \frac{5}{2})\) and \((4, -1)\), let's go through the following steps:

### Step 1: Find the Slope of the Line
The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For points \((-3, \frac{5}{2})\) and \((4, -1)\):
- \( x_1 = -3 \), \( y_1 = \frac{5}{2} \)
- \( x_2 = 4 \), \( y_2 = -1 \)

Calculating the slope:
\[
m = \frac{-1 - \frac{5}{2}}{4 - (-3)} = \frac{-1 - \frac{5}{2}}{4 + 3} = \frac{-\frac{7}{2}}{7} = -\frac{1}{2}
\]

### Step 2: Use the Point-Slope Form to Find the \( y \)-Intercept
The equation of a line in point-slope form is:
\[
y = mx + c
\]
where \( c \) is the \( y \)-intercept.

We can substitute one of the points and the slope to solve for \( c \). Using the point \((4, -1)\) and \( m = -\frac{1}{2} \):
\[
-1 = -\frac{1}{2} \cdot 4 + c
\]
\[
-1 = -2 + c
\]
\[
c = 1
\]

### Step 3: Conclusion
The \( y \)-intercept \( r \) is 1.

### Answer
The correct answer is:
**b. \( 1 \)**

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. What is the general form of a linear equation, and how can it be converted from point-slope form?
2. How can we find the equation of a line if only the slope and one point are given?
3. What is the significance of the \( y \)-intercept in the context of a linear function?
4. How would the process change if we were looking for the \( x \)-intercept instead of the \( y \)-intercept?
5. Can we verify the equation of the line by substituting both points into it?

**Tip:** Always double-check the slope calculation when working with fractions to avoid minor errors that could affect the final answer.

A straight line passes through the points (-3, 5/2) and (4, -1). If the coordinates of the y-intercept of this line are (0, r), then r equals:

Select one:

a. -1/2
b. 1
c. 2

This problem explores how to find the y-intercept (r) of a line passing through two points. The solution includes calculating the slope and using the slope-intercept form of a line equation. Ideal for students in Grades 9-10 learning analytic geometry.

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