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.