Question

write the equation of the line that passes through the points \\((7, -4)\\) and \\((-1, 3)\\), first in point-slope form, and then in slope-intercept form.

the slope of the line is dropdown
when the point \\((7, -4)\\) is used, the point-slope form of the line is dropdown
the slope-intercept form of the line is dropdown

Explanation:

Calculate the slope of the line

We find the slope \(m\) using the coordinates \((x_1, y_1) = (7, -4)\) and \((x_2, y_2) = (-1, 3)\).
Using the slope formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

We apply Variable Substitution to evaluate:

$$m = \frac{3 - (-4)}{-1 - 7} = \frac{7}{-8} = -\frac{7}{8}$$

Write the point-slope form

The point-slope equation is \(y - y_1 = m(x - x_1)\).
Using the point \((7, -4)\) and \(m = -\frac{7}{8}\), we substitute:

$$y - (-4) = -\frac{7}{8}(x - 7)$$

Simplifying the left side gives:

$$y + 4 = -\frac{7}{8}(x - 7)$$

Convert to slope-intercept form

We isolate \(y\) to find the slope-intercept form \(y = mx + b\).
Distribute the slope:

$$y + 4 = -\frac{7}{8}x + \frac{49}{8}$$

Subtract 4 from both sides:

$$y = -\frac{7}{8}x + \frac{49}{8} - \frac{32}{8}$$

Combine the constants:

$$y = -\frac{7}{8}x + \frac{17}{8}$$

Answer:

Write the equation of the line that passes through the points \((7, -4)\) and \((-1, 3)\), first in point-slope form, and then in slope-intercept form.

The slope of the line is <blank>\(-\frac{7}{8}\)</blank>
When the point \((7, -4)\) is used, the point-slope form of the line is <blank>\(y + 4 = -\frac{7}{8}(x - 7)\)</blank>
The slope-intercept form of the line is <blank>\(y = -\frac{7}{8}x + \frac{17}{8}\)</blank>