Question
What is the slope of the line that goes through the points?
x: 0,5,10
y:1300,800,300
x: 0,5,10
y:1300,800,300
Answers
Answered by
GPT-4o mini
To find the slope of the line that goes through the given points \((0, 1300)\), \((5, 800)\), and \((10, 300)\), we can use the formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's calculate the slope between the first two points \((0, 1300)\) and \((5, 800)\):
\[
m = \frac{800 - 1300}{5 - 0} = \frac{-500}{5} = -100
\]
Now, let's calculate the slope between the second two points \((5, 800)\) and \((10, 300)\):
\[
m = \frac{300 - 800}{10 - 5} = \frac{-500}{5} = -100
\]
We see that the slope is constant at \(-100\) for both segments between the points. Therefore, the slope of the line that goes through all three points is:
\[
\boxed{-100}
\]
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's calculate the slope between the first two points \((0, 1300)\) and \((5, 800)\):
\[
m = \frac{800 - 1300}{5 - 0} = \frac{-500}{5} = -100
\]
Now, let's calculate the slope between the second two points \((5, 800)\) and \((10, 300)\):
\[
m = \frac{300 - 800}{10 - 5} = \frac{-500}{5} = -100
\]
We see that the slope is constant at \(-100\) for both segments between the points. Therefore, the slope of the line that goes through all three points is:
\[
\boxed{-100}
\]
Answer
What is the y-intercept of the line that goes through the points?
Answered by
GPT-4o mini
To find the y-intercept of the line that goes through the points \((0, 1300)\), \((5, 800)\), and \((10, 300)\), 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 calculation, we found that the slope \(m\) is \(-100\).
We can use one of the points to solve for the y-intercept \(b\). We'll use the point \((0, 1300)\):
1. Substitute \(x = 0\), \(y = 1300\), and \(m = -100\) into the equation:
\[
1300 = -100(0) + b
\]
This simplifies to:
\[
1300 = b
\]
Thus, the y-intercept is:
\[
\boxed{1300}
\]
This means the line intersects the y-axis at the point \((0, 1300)\).
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.
From the previous calculation, we found that the slope \(m\) is \(-100\).
We can use one of the points to solve for the y-intercept \(b\). We'll use the point \((0, 1300)\):
1. Substitute \(x = 0\), \(y = 1300\), and \(m = -100\) into the equation:
\[
1300 = -100(0) + b
\]
This simplifies to:
\[
1300 = b
\]
Thus, the y-intercept is:
\[
\boxed{1300}
\]
This means the line intersects the y-axis at the point \((0, 1300)\).
Answer
What is the equation of the line that represents the scenario?
Answered by
GPT-4o mini
To find the equation of the line that goes through the points \((0, 1300)\), \((5, 800)\), and \((10, 300)\), we use the slope-intercept form of the equation of a line:
\[
y = mx + b
\]
Where:
- \(m\) is the slope,
- \(b\) is the y-intercept.
From our previous calculations:
- The slope \(m\) is \(-100\).
- The y-intercept \(b\) is \(1300\).
Substituting these values into the slope-intercept equation, we get:
\[
y = -100x + 1300
\]
So the equation of the line that represents the scenario is:
\[
\boxed{y = -100x + 1300}
\]
\[
y = mx + b
\]
Where:
- \(m\) is the slope,
- \(b\) is the y-intercept.
From our previous calculations:
- The slope \(m\) is \(-100\).
- The y-intercept \(b\) is \(1300\).
Substituting these values into the slope-intercept equation, we get:
\[
y = -100x + 1300
\]
So the equation of the line that represents the scenario is:
\[
\boxed{y = -100x + 1300}
\]