Sure! To find the equation of the line passing through the two given points, [tex]\((2,-6)\)[/tex] and [tex]\((-4,-4)\)[/tex], let's follow these steps:
1. Calculate the slope ([tex]\(m\)[/tex]) of the line:
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((2, -6)\)[/tex] and [tex]\((-4, -4)\)[/tex]:
[tex]\[ m = \frac{-4 - (-6)}{-4 - 2} = \frac{-4 + 6}{-4 - 2} = \frac{2}{-6} = -\frac{1}{3} \][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(-\frac{1}{3}\)[/tex].
2. Find the y-intercept ([tex]\(b\)[/tex]) of the line:
Using the slope-intercept form of the equation of a line, which is [tex]\(y = mx + b\)[/tex], we need to solve for [tex]\(b\)[/tex]. We can use one of the given points to do this. Let's use the point [tex]\((2, -6)\)[/tex].
Substitute [tex]\(m = -\frac{1}{3}\)[/tex], [tex]\(x = 2\)[/tex], and [tex]\(y = -6\)[/tex] into the equation [tex]\(y = mx + b\)[/tex]:
[tex]\[ -6 = -\frac{1}{3}(2) + b \][/tex]
Simplify and solve for [tex]\(b\)[/tex]:
[tex]\[ -6 = -\frac{2}{3} + b \][/tex]
To isolate [tex]\(b\)[/tex], add [tex]\(\frac{2}{3}\)[/tex] to both sides:
[tex]\[ -6 + \frac{2}{3} = b \][/tex]
Convert [tex]\(-6\)[/tex] to a fraction with a common denominator:
[tex]\[ -6 = -\frac{18}{3} \][/tex]
Thus:
[tex]\[ -\frac{18}{3} + \frac{2}{3} = b \][/tex]
[tex]\[ b = -\frac{16}{3} \][/tex]
So, the y-intercept [tex]\(b\)[/tex] is [tex]\(-\frac{16}{3}\)[/tex].
3. Write the equation of the line:
Now that we have the slope [tex]\(m = -\frac{1}{3}\)[/tex] and the y-intercept [tex]\(b = -\frac{16}{3}\)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[ y = -\frac{1}{3}x - \frac{16}{3} \][/tex]
Therefore, the equation of the line that passes through the points [tex]\((2, -6)\)[/tex] and [tex]\((-4, -4)\)[/tex] is:
[tex]\[ y = -\frac{1}{3}x - \frac{16}{3} \][/tex]
