To find the equation of the line passing through the given points [tex]\((2, -8)\)[/tex] and [tex]\((-1, 7)\)[/tex], we need to determine the slope of the line and the y-intercept. Once we have these, we can write the equation in the slope-intercept form [tex]\(y = mx + b\)[/tex].
### Step 1: Calculate the slope (m)
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]
Substituting the given points [tex]\((2, -8)\)[/tex] and [tex]\((-1, 7)\)[/tex]:
[tex]\[
m = \frac{7 - (-8)}{-1 - 2} = \frac{7 + 8}{-1 - 2} = \frac{15}{-3} = -5
\][/tex]
So, the slope of the line is [tex]\(m = -5\)[/tex].
### Step 2: Calculate the y-intercept (b)
To find the y-intercept, we can use the slope-intercept form of the equation [tex]\(y = mx + b\)[/tex] and one of the given points. We'll use the point [tex]\((2, -8)\)[/tex].
Substitute [tex]\(m = -5\)[/tex], [tex]\(x = 2\)[/tex], and [tex]\(y = -8\)[/tex]:
[tex]\[
-8 = -5(2) + b
\][/tex]
[tex]\[
-8 = -10 + b
\][/tex]
Adding 10 to both sides to solve for [tex]\(b\)[/tex]:
[tex]\[
-8 + 10 = b
\][/tex]
[tex]\[
b = 2
\][/tex]
So, the y-intercept is [tex]\(b = 2\)[/tex].
### Step 3: Write the equation of the line
Now that we have the slope [tex]\(m = -5\)[/tex] and the y-intercept [tex]\(b = 2\)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[
y = -5x + 2
\][/tex]
Thus, the equation of the line passing through the points [tex]\((2, -8)\)[/tex] and [tex]\((-1, 7)\)[/tex] is:
[tex]\[
y = -5x + 2
\][/tex]
