Question: Determine an equation of the line that passes through the points $(-2, 10)$ and $(6, 4)$. Write your answer in the form $ax + by = c$, where $a$, $b$ and $c$ are integers in their lowest terms.
1. **State the problem:** Find the equation of the line passing through points $(-2, 10)$ and $(6, 4)$ in the form $ax + by = c$ with integers $a$, $b$, and $c$ in lowest terms.
2. **Formula used:** The slope $m$ of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
The point-slope form of the line is:
$$y - y_1 = m(x - x_1)$$
We will then convert this to standard form $ax + by = c$.
3. **Calculate the slope:**
$$m = \frac{4 - 10}{6 - (-2)} = \frac{-6}{8} = \frac{-3}{4}$$
4. **Use point-slope form with point $(-2, 10)$:**
$$y - 10 = -\frac{3}{4}(x - (-2))$$
$$y - 10 = -\frac{3}{4}(x + 2)$$
5. **Distribute the slope:**
$$y - 10 = -\frac{3}{4}x - \frac{3}{4} \times 2$$
$$y - 10 = -\frac{3}{4}x - \frac{3}{2}$$
6. **Add 10 to both sides:**
$$y = -\frac{3}{4}x - \frac{3}{2} + 10$$
$$y = -\frac{3}{4}x + \frac{17}{2}$$
7. **Convert to standard form $ax + by = c$:**
Multiply both sides by 4 to clear denominators:
$$4y = -3x + 34$$
Rewrite:
$$3x + 4y = 34$$
8. **Check for lowest terms:**
Coefficients $3$, $4$, and $34$ have no common divisor other than 1, so this is the simplest form.
**Final answer:**
$$3x + 4y = 34$$