To find the slope-intercept equation of the line that goes through the points (2, -10) and (8, -16), we need to follow these steps:
### Step 1: Calculate the Slope
The slope [tex]\( m \)[/tex] of a line passing through 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 given points into the formula:
[tex]\[ x_1 = 2, \, y_1 = -10, \, x_2 = 8, \, y_2 = -16 \][/tex]
[tex]\[ m = \frac{-16 - (-10)}{8 - 2} \][/tex]
[tex]\[ m = \frac{-16 + 10}{8 - 2} \][/tex]
[tex]\[ m = \frac{-6}{6} \][/tex]
[tex]\[ m = -1 \][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\(-1\)[/tex].
### Step 2: Find the Y-intercept
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. To find [tex]\( b \)[/tex], we use one of the points and solve for [tex]\( b \)[/tex]. Let's use the point [tex]\((2, -10)\)[/tex].
Substitute [tex]\( m = -1 \)[/tex] and [tex]\((x_1, y_1) = (2, -10) \)[/tex] into the equation [tex]\( y = mx + b \)[/tex]:
[tex]\[ -10 = (-1)(2) + b \][/tex]
[tex]\[ -10 = -2 + b \][/tex]
[tex]\[ b = -10 + 2 \][/tex]
[tex]\[ b = -8 \][/tex]
So, the y-intercept [tex]\( b \)[/tex] is [tex]\(-8\)[/tex].
### Step 3: Write the Final Equation
Now that we have the slope [tex]\( m = -1 \)[/tex] and the y-intercept [tex]\( b = -8 \)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[ y = -1x - 8 \][/tex]
or simply
[tex]\[ y = -x - 8 \][/tex]
So, the equation of the line that goes through the points (2, -10) and (8, -16) is:
[tex]\[ y = -x - 8 \][/tex]
