To find the equation of the line passing through two points, we use the point-slope form of a line:

$y - y_1 = m(x - x_1),$ where $m$ is the slope, and $(x_1, y_1)$ is a point on the line.

Step 1: Find the slope ($m$)

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 points $(3, -4)$ and $(-4, 3)$: $m = \frac{3 - (-4)}{-4 - 3} = \frac{3 + 4}{-7} = \frac{7}{-7} = -1.$

Step 2: Use the point-slope form

Choose one of the points, say $(3, -4)$. Substitute $m = -1$, $x_1 = 3$, and $y_1 = -4$ into the point-slope form: $y - (-4) = -1(x - 3).$

Simplify: $y + 4 = -x + 3.$

Step 3: Rearrange to slope-intercept form ($y = mx + b$):

$y = -x + 3 - 4,$ $y = -x - 1.$

Final Answer:

The equation of the line is: $y = -x - 1.$

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Related Questions:

1. How can the slope-intercept form be converted to standard form?
2. How do parallel lines relate to their slopes?
3. What would be the equation of a line perpendicular to this one?
4. How can you check if a point lies on a given line?
5. What is the distance between the two points given in this problem?

Tip:

The slope is a measure of steepness, and its sign indicates direction (positive for upward, negative for downward).