Question
Find the slope of the line that passes through the given points. Then determine the Vertical Intercept. Use the Slope and Vertical Intercept to write the Equation for the Line. The first one is done for you as an example.
Note: If the slope does not exist, enter DNE
ordered pairs (-8,24) and (4,-12)
slope
verticle intercept
equation of the line
Note: If the slope does not exist, enter DNE
ordered pairs (-8,24) and (4,-12)
slope
verticle intercept
equation of the line
Answers
GPT-4o mini
To find the slope, vertical intercept, and the equation of the line that passes through the points (-8, 24) and (4, -12), we'll follow these steps:
### Step 1: Calculate the slope
The slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the given points \((-8, 24)\) and \((4, -12)\):
- \((x_1, y_1) = (-8, 24)\)
- \((x_2, y_2) = (4, -12)\)
\[
m = \frac{-12 - 24}{4 - (-8)} = \frac{-36}{4 + 8} = \frac{-36}{12} = -3
\]
### Step 2: Determine the vertical intercept
To find the vertical intercept (y-intercept) \( b \) of the line, we can use the point-slope form of the equation of a line, which is given by:
\[
y = mx + b
\]
We can use either of the given points to solve for \( b \). Using the point \((4, -12)\):
\[
-12 = -3(4) + b
\]
Calculating:
\[
-12 = -12 + b
\]
Adding 12 to both sides:
\[
b = 0
\]
### Step 3: Write the equation of the line
Now that we have the slope \( m = -3 \) and the vertical intercept \( b = 0 \), we can write the equation of the line in slope-intercept form \( y = mx + b \):
\[
y = -3x + 0
\]
Simplifying this gives:
\[
y = -3x
\]
### Final Summary
- **Slope**: \(-3\)
- **Vertical Intercept**: \(0\)
- **Equation of the Line**: \(y = -3x\)
### Step 1: Calculate the slope
The slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the given points \((-8, 24)\) and \((4, -12)\):
- \((x_1, y_1) = (-8, 24)\)
- \((x_2, y_2) = (4, -12)\)
\[
m = \frac{-12 - 24}{4 - (-8)} = \frac{-36}{4 + 8} = \frac{-36}{12} = -3
\]
### Step 2: Determine the vertical intercept
To find the vertical intercept (y-intercept) \( b \) of the line, we can use the point-slope form of the equation of a line, which is given by:
\[
y = mx + b
\]
We can use either of the given points to solve for \( b \). Using the point \((4, -12)\):
\[
-12 = -3(4) + b
\]
Calculating:
\[
-12 = -12 + b
\]
Adding 12 to both sides:
\[
b = 0
\]
### Step 3: Write the equation of the line
Now that we have the slope \( m = -3 \) and the vertical intercept \( b = 0 \), we can write the equation of the line in slope-intercept form \( y = mx + b \):
\[
y = -3x + 0
\]
Simplifying this gives:
\[
y = -3x
\]
### Final Summary
- **Slope**: \(-3\)
- **Vertical Intercept**: \(0\)
- **Equation of the Line**: \(y = -3x\)