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