To find the equation of the line passing through the points [tex]\((7, -4)\)[/tex] and [tex]\((-1, 2)\)[/tex], we need to determine the slope-intercept form of the line, which is given by [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
### Step 1: Calculate the Slope (m)
The slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates of the given points [tex]\((7, -4)\)[/tex] and [tex]\((-1, 2)\)[/tex]:
[tex]\[ m = \frac{2 - (-4)}{-1 - 7} \][/tex]
[tex]\[ m = \frac{2 + 4}{-1 - 7} \][/tex]
[tex]\[ m = \frac{6}{-8} \][/tex]
[tex]\[ m = -\frac{3}{4} \][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(-\frac{3}{4}\)[/tex].
### Step 2: Determine the y-Intercept (b)
Now that we have the slope, we use one of the points to find the y-intercept [tex]\(b\)[/tex]. We'll use the point [tex]\((7, -4)\)[/tex] and the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ -4 = -\frac{3}{4} \cdot 7 + b \][/tex]
Solve for [tex]\(b\)[/tex]:
[tex]\[ -4 = -\frac{3}{4} \cdot 7 + b \][/tex]
[tex]\[ -4 = -\frac{21}{4} + b \][/tex]
[tex]\[ -4 + \frac{21}{4} = b \][/tex]
[tex]\[ b = -\frac{16}{4} + \frac{21}{4} \][/tex]
[tex]\[ b = \frac{5}{4} \][/tex]
So, the y-intercept [tex]\(b\)[/tex] is [tex]\(\frac{5}{4}\)[/tex].
### Step 3: Write the Equation of the Line
With the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex], we can write the equation of the line:
[tex]\[ y = -\frac{3}{4}x + \frac{5}{4} \][/tex]
Therefore, the correct equation of the line passing through the points [tex]\((7, -4)\)[/tex] and [tex]\((-1, 2)\)[/tex] in slope-intercept form is:
c. [tex]\( y = -\frac{3}{4}x + \frac{5}{4} \)[/tex]
