Question: User: where did you get 36/5 from in point number 4
State the problem: We need to find the equation of the line that passes through the points $(3,-5)$ and $(-2,7)$.
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.
Find the slope $m$: The slope formula is:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Using the points $(3,-5)$ and $(-2,7)$:
$$m = \frac{7 - (-5)}{-2 - 3} = \frac{7 + 5}{-5} = \frac{12}{-5} = -\frac{12}{5}$$
Use point-slope form: Choose point $(3,-5)$:
$$y - (-5) = -\frac{12}{5}(x - 3)$$
which simplifies to:
$$y + 5 = -\frac{12}{5}x + -\frac{12}{5} \times (-3)$$
Calculate the product:
$$-\frac{12}{5} \times (-3) = \frac{36}{5}$$
So:
$$y + 5 = -\frac{12}{5}x + \frac{36}{5}$$
Isolate $y$ to get slope-intercept form:
$$y = -\frac{12}{5}x + \frac{36}{5} - 5$$
Rewrite $5$ as $\frac{25}{5}$:
$$y = -\frac{12}{5}x + \frac{36}{5} - \frac{25}{5}$$
Simplify constants:
$$y = -\frac{12}{5}x + \frac{11}{5}$$
Final answer: The equation of the line is
$$y = -\frac{12}{5}x + \frac{11}{5}$$
The $\frac{36}{5}$ comes from multiplying the slope $-\frac{12}{5}$ by $-3$ in the point-slope formula step.
1. **State the problem:** Find the equation of the line passing through points $(3,-5)$ and $(-2,7)$.
2. **Formula used:** The point-slope form of a line is:
$$y - y_1 = m(x - x_1)$$
where $m$ is the slope and $(x_1,y_1)$ is a point on the line.
3. **Find the slope $m$:**
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - (-5)}{-2 - 3} = \frac{7 + 5}{-5} = \frac{12}{-5} = -\frac{12}{5}$$
4. **Use point-slope form with point $(3,-5)$:**
$$y - (-5) = -\frac{12}{5}(x - 3)$$
which is:
$$y + 5 = -\frac{12}{5}x + -\frac{12}{5} \times (-3)$$
5. **Calculate $-\frac{12}{5} \times (-3)$:**
$$-\frac{12}{5} \times (-3) = \frac{36}{5}$$
This is where $\frac{36}{5}$ comes from.
6. **Rewrite the equation:**
$$y + 5 = -\frac{12}{5}x + \frac{36}{5}$$
7. **Isolate $y$:**
$$y = -\frac{12}{5}x + \frac{36}{5} - 5$$
Rewrite $5$ as $\frac{25}{5}$:
$$y = -\frac{12}{5}x + \frac{36}{5} - \frac{25}{5}$$
8. **Simplify constants:**
$$y = -\frac{12}{5}x + \frac{11}{5}$$
9. **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.