Question: 6) Find the equation of the line joining the points $(3, -5)$ and $(-2, 7)$.
1. **State the problem:** We need to find the equation of the line that passes through the points $(3, -5)$ and $(-2, 7)$.
2. **Formula used:** The equation of a line can be found using the point-slope form:
$$y - y_1 = m(x - x_1)$$
where $m$ is the slope of the line and $(x_1, y_1)$ is a point on the line.
3. **Find the slope $m$:** The slope formula is:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Using the points $(3, -5)$ and $(-2, 7)$:
$$m = \frac{7 - (-5)}{-2 - 3} = \frac{7 + 5}{-5} = \frac{12}{-5} = -\frac{12}{5}$$
4. **Use point-slope form:** Choose point $(3, -5)$:
$$y - (-5) = -\frac{12}{5}(x - 3)$$
which simplifies to:
$$y + 5 = -\frac{12}{5}x + \frac{36}{5}$$
5. **Isolate $y$ to get slope-intercept form:**
$$y = -\frac{12}{5}x + \frac{36}{5} - 5$$
Rewrite $5$ as $\frac{25}{5}$:
$$y = -\frac{12}{5}x + \frac{36}{5} - \frac{25}{5}$$
6. **Simplify constants:**
$$y = -\frac{12}{5}x + \frac{11}{5}$$
**Final answer:** The equation of the line is
$$y = -\frac{12}{5}x + \frac{11}{5}$$
This means for any $x$, you can find $y$ on the line using this formula.