Find the equation of the line that passes through the points (1, - 4) and (- 3, 2).

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.

Learn how to find the equation of the line passing through points (1, -4) and (-3, 2) using the slope formula and point-slope form. This detailed solution covers essential algebra and geometry concepts, suitable for students in Grades 8-10.

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