To find the equation of the line passing through the points [tex]\((-1, 1)\)[/tex], [tex]\((4, 0)\)[/tex], and [tex]\((24, -4)\)[/tex], we need to determine if these points are collinear and then derive the equation of the line. Here is a detailed, step-by-step solution:
### Step 1: Calculate the slope between the first two points [tex]\((-1, 1)\)[/tex] and [tex]\((4, 0)\)[/tex]
The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((-1, 1)\)[/tex] and [tex]\((4, 0)\)[/tex]:
[tex]\[ m = \frac{0 - 1}{4 - (-1)} = \frac{0 - 1}{4 + 1} = \frac{-1}{5} = -\frac{1}{5} \][/tex]
So, the slope [tex]\(m_1\)[/tex] between [tex]\((-1, 1)\)[/tex] and [tex]\((4, 0)\)[/tex] is [tex]\(-\frac{1}{5}\)[/tex].
### Step 2: Calculate the slope between the next two points [tex]\((4, 0)\)[/tex] and [tex]\((24, -4)\)[/tex]
Using the same slope formula for the points [tex]\((4, 0)\)[/tex] and [tex]\((24, -4)\)[/tex]:
[tex]\[ m = \frac{-4 - 0}{24 - 4} = \frac{-4}{20} = -\frac{1}{5} \][/tex]
So, the slope [tex]\(m_2\)[/tex] between [tex]\((4, 0)\)[/tex] and [tex]\((24, -4)\)[/tex] is also [tex]\(-\frac{1}{5}\)[/tex].
### Step 3: Check if the points are collinear
Since both slopes [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex] are equal ([tex]\(-\frac{1}{5}\)[/tex]), the points are collinear. This means they lie on the same straight line.
### Step 4: Determine the equation of the line
Since the points are collinear and the slope of the line is [tex]\(-\frac{1}{5}\)[/tex], we can use the point-slope form of the line equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using the point [tex]\((-1, 1)\)[/tex] and the slope [tex]\(m = -\frac{1}{5}\)[/tex]:
[tex]\[ y - 1 = -\frac{1}{5}(x - (-1)) \][/tex]
[tex]\[ y - 1 = -\frac{1}{5}(x + 1) \][/tex]
[tex]\[ y - 1 = -\frac{1}{5}x - \frac{1}{5} \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{1}{5}x - \frac{1}{5} + 1 \][/tex]
[tex]\[ y = -\frac{1}{5}x + \frac{4}{5} \][/tex]
Thus, the equation of the line is:
[tex]\[ y = -\frac{1}{5}x + \frac{4}{5} \][/tex]
### Conclusion
Among the given options, the equation that matches this derivation is:
[tex]\[ y = -\frac{1}{5} x + \frac{4}{5} \][/tex]
Therefore, the correct equation of the line through the points [tex]\((-1, 1)\)[/tex], [tex]\((4, 0)\)[/tex], and [tex]\((24, -4)\)[/tex] is [tex]\(y = -\frac{1}{5} x + \frac{4}{5}\)[/tex], corresponding to option 2.
