To find the equation of the line passing through the points $(1, -4)$ and $(-3, 2)$, we follow these steps:

1. Find the slope (m) of the line:

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute the given points $(1, -4)$ and $(-3, 2)$ into the formula:

$m = \frac{2 - (-4)}{-3 - 1} = \frac{2 + 4}{-3 - 1} = \frac{6}{-4} = -\frac{3}{2}$

2. Use the point-slope form of the equation of a line:

The point-slope form is: $y - y_1 = m(x - x_1)$ Choose one of the points, say $(1, -4)$, and substitute it into the formula along with the slope $m = -\frac{3}{2}$:

$y - (-4) = -\frac{3}{2}(x - 1)$ This simplifies to: $y + 4 = -\frac{3}{2}(x - 1)$

3. Simplify to get the slope-intercept form:

Now, expand the right side: $y + 4 = -\frac{3}{2}x + \frac{3}{2}$

Subtract 4 from both sides: $y = -\frac{3}{2}x + \frac{3}{2} - 4$

Convert 4 to a fraction and simplify: $y = -\frac{3}{2}x + \frac{3}{2} - \frac{8}{2}$ $y = -\frac{3}{2}x - \frac{5}{2}$

Thus, the equation of the line is: $y = -\frac{3}{2}x - \frac{5}{2}$

Would you like further clarification or details?

Related Questions:

1. How do you convert the equation from point-slope to slope-intercept form?
2. What is the general formula for finding the slope of a line between two points?
3. How can we determine the x-intercept and y-intercept of the line equation?
4. How would the equation change if the line passed through different points?
5. What is the geometric interpretation of the slope $m = -\frac{3}{2}$?

Tip:

When simplifying fractions involving negative signs, always double-check that the operations align with the rules for addition and subtraction to avoid sign errors.