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Let's solve this step-by-step.

We have two points: \( A(-1, -2) \) and \( B(3, 5) \). We need to find:
1. **The equation of the line passing through these points.**
2. **The slope of this line.**
3. **The x- and y-intercepts of the line.**

### Step (a): Find the Equation of the Line
To find the equation of the line, we start with the slope-intercept form:
\[
y = mx + c
\]
where \( m \) is the slope and \( c \) is the y-intercept.

#### Calculate the Slope \( m \):
The slope \( m \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substitute \( (x_1, y_1) = (-1, -2) \) and \( (x_2, y_2) = (3, 5) \):
\[
m = \frac{5 - (-2)}{3 - (-1)} = \frac{5 + 2}{3 + 1} = \frac{7}{4}
\]

So, the slope \( m \) is \( \frac{7}{4} \).

#### Form the Equation Using Point-Slope Form
Using the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), and choosing point \( A(-1, -2) \):
\[
y - (-2) = \frac{7}{4}(x - (-1))
\]
\[
y + 2 = \frac{7}{4}(x + 1)
\]
Expanding this:
\[
y + 2 = \frac{7}{4}x + \frac{7}{4}
\]
\[
y = \frac{7}{4}x + \frac{7}{4} - 2
\]
Converting \( -2 \) to \( \frac{-8}{4} \):
\[
y = \frac{7}{4}x - \frac{1}{4}
\]

Thus, the equation of the line is:
\[
y = \frac{7}{4}x - \frac{1}{4}
\]

### Step (b): Find the Slope
The slope \( m \) was calculated as \( \frac{7}{4} \) in the first part.

### Step (c): Find the x- and y-intercepts
#### x-intercept:
The x-intercept occurs when \( y = 0 \):
\[
0 = \frac{7}{4}x - \frac{1}{4}
\]
Multiply both sides by 4:
\[
0 = 7x - 1
\]
\[
7x = 1
\]
\[
x = \frac{1}{7}
\]

So, the x-intercept is \( \left( \frac{1}{7}, 0 \right) \).

#### y-intercept:
The y-intercept occurs when \( x = 0 \):
\[
y = \frac{7}{4}(0) - \frac{1}{4} = -\frac{1}{4}
\]

So, the y-intercept is \( \left( 0, -\frac{1}{4} \right) \).

### Summary:
1. **Equation of the line:** \( y = \frac{7}{4}x - \frac{1}{4} \)
2. **Slope:** \( \frac{7}{4} \)
3. **x-intercept:** \( \left( \frac{1}{7}, 0 \right) \)
4. **y-intercept:** \( \left( 0, -\frac{1}{4} \right) \)

Would you like further details on any part of this solution?

#### Related Questions:
1. How do you convert a slope-intercept form to standard form?
2. What are the implications of a positive or negative slope in a line equation?
3. How can you find the distance between two points on a coordinate plane?
4. How do you determine if two lines are parallel or perpendicular?
5. How would you derive the equation of a line given its x- and y-intercepts?

#### Tip:
When working with points and lines, drawing a quick sketch can often help visualize the slope and intercepts, making it easier to confirm your calculations.

A(-1, -2) ve B(3,5) noktalarından geçen doğrunun
a) denklemini bulunuz.
b) eğimini bulunuz.
c) x ve y eksenlerini kestiği noktaları bulunuz.

Learn how to find the equation, slope, and intercepts of a line passing through points A(-1, -2) and B(3, 5). This solution provides a step-by-step guide on using the slope formula and point-slope form, suitable for Grades 9-10.

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