Question
What is the slope of the line that goes through the points in the table?
x: 0, 2,4,6,8
y: -1,-2,-3,-4,5
x: 0, 2,4,6,8
y: -1,-2,-3,-4,5
Answers
Answered by
GPT-4o mini
To find the slope of the line that goes through the points in the table, we can use the formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We can select any two points from the table. Let's take the first two points:
- Point 1: \((0, -1)\)
- Point 2: \((2, -2)\)
Using these points:
\[
m = \frac{-2 - (-1)}{2 - 0} = \frac{-2 + 1}{2} = \frac{-1}{2}
\]
Now, let's calculate the slope using another pair of points to verify consistency. We can use the points \((4, -3)\) and \((6, -4)\):
- Point 3: \((4, -3)\)
- Point 4: \((6, -4)\)
Now, calculating the slope:
\[
m = \frac{-4 - (-3)}{6 - 4} = \frac{-4 + 3}{2} = \frac{-1}{2}
\]
The slope calculated is consistent for these points. Hence, the slope of the line that goes through the points in the table is:
\[
\boxed{-\frac{1}{2}}
\]
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We can select any two points from the table. Let's take the first two points:
- Point 1: \((0, -1)\)
- Point 2: \((2, -2)\)
Using these points:
\[
m = \frac{-2 - (-1)}{2 - 0} = \frac{-2 + 1}{2} = \frac{-1}{2}
\]
Now, let's calculate the slope using another pair of points to verify consistency. We can use the points \((4, -3)\) and \((6, -4)\):
- Point 3: \((4, -3)\)
- Point 4: \((6, -4)\)
Now, calculating the slope:
\[
m = \frac{-4 - (-3)}{6 - 4} = \frac{-4 + 3}{2} = \frac{-1}{2}
\]
The slope calculated is consistent for these points. Hence, the slope of the line that goes through the points in the table is:
\[
\boxed{-\frac{1}{2}}
\]
Answer
What is the y-intercept of the line that goes through the points in the table?
Answered by
GPT-4o mini
To find the y-intercept of the line that goes through the points in the table, we can use the slope-intercept form of the equation of a line, which is:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
From the previous calculations, we found the slope \( m = -\frac{1}{2} \). Now we need to find \( b \) by plugging in one of the points from the table into the equation. Let's use the point \((0, -1)\):
1. Substitute \( x = 0 \) and \( y = -1 \) into the equation:
\[
-1 = -\frac{1}{2}(0) + b
\]
2. This simplifies to:
\[
-1 = b
\]
Therefore, the y-intercept \( b \) is:
\[
\boxed{-1}
\]
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
From the previous calculations, we found the slope \( m = -\frac{1}{2} \). Now we need to find \( b \) by plugging in one of the points from the table into the equation. Let's use the point \((0, -1)\):
1. Substitute \( x = 0 \) and \( y = -1 \) into the equation:
\[
-1 = -\frac{1}{2}(0) + b
\]
2. This simplifies to:
\[
-1 = b
\]
Therefore, the y-intercept \( b \) is:
\[
\boxed{-1}
\]
Answer
What's the equation of the line that goes through all of the points in the table in slope-intercept form?
Answered by
GPT-4o mini
The equation of a line in slope-intercept form is given by:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
From our earlier calculations:
- The slope \( m \) is \(-\frac{1}{2}\).
- The y-intercept \( b \) is \(-1\).
Now we can plug these values into the slope-intercept form to find the equation of the line.
\[
y = -\frac{1}{2}x - 1
\]
Thus, the equation of the line that goes through all of the points in the table in slope-intercept form is:
\[
\boxed{y = -\frac{1}{2}x - 1}
\]
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
From our earlier calculations:
- The slope \( m \) is \(-\frac{1}{2}\).
- The y-intercept \( b \) is \(-1\).
Now we can plug these values into the slope-intercept form to find the equation of the line.
\[
y = -\frac{1}{2}x - 1
\]
Thus, the equation of the line that goes through all of the points in the table in slope-intercept form is:
\[
\boxed{y = -\frac{1}{2}x - 1}
\]