To find the equation of the line passing through the points $(7, -4)$ and $(-1, 3)$, we’ll first calculate the slope and then use point-slope form.

Step 1: Calculate the Slope

The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting $(x_1, y_1) = (7, -4)$ and $(x_2, y_2) = (-1, 3)$: $m = \frac{3 - (-4)}{-1 - 7} = \frac{3 + 4}{-8} = \frac{7}{-8} = -\frac{7}{8}$

Step 2: Write in Point-Slope Form

The point-slope form of a line with slope $m$ passing through a point $(x_1, y_1)$ is: $y - y_1 = m(x - x_1)$ Using the point $(7, -4)$ and the slope $-\frac{7}{8}$: $y + 4 = -\frac{7}{8}(x - 7)$ This is the point-slope form of the line.

Step 3: Convert to Slope-Intercept Form

Now, let’s simplify this equation to the slope-intercept form, $y = mx + b$: $y + 4 = -\frac{7}{8}x + \frac{49}{8}$ Subtract 4 from both sides to isolate $y$: $y = -\frac{7}{8}x + \frac{49}{8} - 4$ To combine terms, convert 4 to a fraction with a denominator of 8: $y = -\frac{7}{8}x + \frac{49}{8} - \frac{32}{8}$ $y = -\frac{7}{8}x + \frac{17}{8}$

Final Answer

• Point-Slope Form: $y + 4 = -\frac{7}{8}(x - 7)$
• Slope-Intercept Form: $y = -\frac{7}{8}x + \frac{17}{8}$

Would you like further details or have any questions?

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Here are five related questions for practice:

1. What is the equation of a line passing through points $(3, 2)$ and $(6, -1)$?
2. How do you find the y-intercept of a line given two points?
3. How can you convert a point-slope form equation to standard form?
4. What does the slope of a line tell you about the line's direction?
5. How would you find the x-intercept of the line we just found?

Tip: When finding the slope, always subtract consistently in the same order to avoid sign errors.