To find the equation of the line that passes through the points [tex]\((-2, 8)\)[/tex] and [tex]\( (1, -1) \)[/tex], we begin by determining the slope ([tex]\(m\)[/tex]) and the y-intercept ([tex]\(b\)[/tex]).
### Step 1: Calculate the Slope ([tex]\(m\)[/tex])
The slope [tex]\(m\)[/tex] of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Given the points [tex]\((-2, 8)\)[/tex] and [tex]\((1, -1)\)[/tex]:
[tex]\[ x_1 = -2, \, y_1 = 8 \][/tex]
[tex]\[ x_2 = 1, \, y_2 = -1 \][/tex]
Substituting these values into the slope formula:
[tex]\[ m = \frac{-1 - 8}{1 - (-2)} = \frac{-9}{1 + 2} = \frac{-9}{3} = -3.0 \][/tex]
So, the slope [tex]\(m\)[/tex] of the line is [tex]\(-3.0\)[/tex].
### Step 2: Calculate the Y-Intercept ([tex]\(b\)[/tex])
The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
To find [tex]\(b\)[/tex], we can use one of the points and solve for [tex]\(b\)[/tex]. Let's use the point [tex]\((-2, 8)\)[/tex]:
[tex]\[ y = -3.0x + b \][/tex]
Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = 8\)[/tex] into the equation:
[tex]\[ 8 = -3.0(-2) + b \][/tex]
[tex]\[ 8 = 6 + b \][/tex]
[tex]\[ b = 8 - 6 \][/tex]
[tex]\[ b = 2.0 \][/tex]
### Step 3: Write the Equation of the Line
Now that we have [tex]\(m = -3.0\)[/tex] and [tex]\(b = 2.0\)[/tex], we can write the equation of the line:
[tex]\[ y = -3.0x + 2.0 \][/tex]
Thus, the equation of the line passing through the points [tex]\((-2, 8)\)[/tex] and [tex]\((1, -1)\)[/tex] is:
[tex]\[ y = -3.0x + 2.0 \][/tex]
