Question
The table shows a linear relationship between the variables x and y
x y
5 16.5
7 21.5
11 31.5
What are the slope and y - intercept of the linear relationship
answers:
slope 2, y - intercept (0 , 6.5)
slope -2 , y - intercept (4 , 0)
slope -2.5, y intercept (4 , 0)
slope 2.5, y - intercept (0 , 4)
x y
5 16.5
7 21.5
11 31.5
What are the slope and y - intercept of the linear relationship
answers:
slope 2, y - intercept (0 , 6.5)
slope -2 , y - intercept (4 , 0)
slope -2.5, y intercept (4 , 0)
slope 2.5, y - intercept (0 , 4)
Answers
GPT-4o mini
To find the slope and y-intercept of the linear relationship between the variables \(x\) and \(y\), we can use the points provided.
1. **Calculating the slope (m)**:
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's use the points \( (5, 16.5) \) and \( (7, 21.5) \) to calculate the slope:
\[
m = \frac{21.5 - 16.5}{7 - 5} = \frac{5}{2} = 2.5
\]
Now, let's confirm this by using a different pair of points, \( (7, 21.5) \) and \( (11, 31.5) \):
\[
m = \frac{31.5 - 21.5}{11 - 7} = \frac{10}{4} = 2.5
\]
Thus, the slope is consistent and equals \(2.5\).
2. **Finding the y-intercept (b)**:
To find the y-intercept, we can use the slope-intercept form of the equation of a line, which is given by:
\[
y = mx + b
\]
We can use one of the points, say \( (5, 16.5) \), for this:
\[
16.5 = 2.5(5) + b
\]
Calculating:
\[
16.5 = 12.5 + b
\]
\[
b = 16.5 - 12.5 = 4
\]
So, the y-intercept is \( 4 \).
Now, summarizing the results:
- Slope: \( 2.5 \)
- Y-intercept: \( (0, 4) \)
Therefore, the correct answer is:
**slope 2.5, y - intercept (0 , 4)**.
1. **Calculating the slope (m)**:
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's use the points \( (5, 16.5) \) and \( (7, 21.5) \) to calculate the slope:
\[
m = \frac{21.5 - 16.5}{7 - 5} = \frac{5}{2} = 2.5
\]
Now, let's confirm this by using a different pair of points, \( (7, 21.5) \) and \( (11, 31.5) \):
\[
m = \frac{31.5 - 21.5}{11 - 7} = \frac{10}{4} = 2.5
\]
Thus, the slope is consistent and equals \(2.5\).
2. **Finding the y-intercept (b)**:
To find the y-intercept, we can use the slope-intercept form of the equation of a line, which is given by:
\[
y = mx + b
\]
We can use one of the points, say \( (5, 16.5) \), for this:
\[
16.5 = 2.5(5) + b
\]
Calculating:
\[
16.5 = 12.5 + b
\]
\[
b = 16.5 - 12.5 = 4
\]
So, the y-intercept is \( 4 \).
Now, summarizing the results:
- Slope: \( 2.5 \)
- Y-intercept: \( (0, 4) \)
Therefore, the correct answer is:
**slope 2.5, y - intercept (0 , 4)**.
you sure
GPT-4o mini
Yes, I am confident in the calculations. Let's verify the steps again briefly.
1. **Calculate the slope (m)**:
Using the points \((5, 16.5)\) and \((7, 21.5)\):
\[
m = \frac{21.5 - 16.5}{7 - 5} = \frac{5}{2} = 2.5
\]
Using another pair \((7, 21.5)\) and \((11, 31.5)\):
\[
m = \frac{31.5 - 21.5}{11 - 7} = \frac{10}{4} = 2.5
\]
Both calculations confirm that the slope is \(2.5\).
2. **Calculate the y-intercept (b)**:
Using the slope-intercept form \(y = mx + b\) with the point \((5, 16.5)\):
\[
16.5 = 2.5(5) + b
\]
\[
16.5 = 12.5 + b
\]
\[
b = 16.5 - 12.5 = 4
\]
Thus, the y-intercept is \(4\).
Therefore, the slope is \(2.5\) and the y-intercept is \((0, 4)\).
The final answer is indeed:
**slope 2.5, y - intercept (0 , 4)**.
1. **Calculate the slope (m)**:
Using the points \((5, 16.5)\) and \((7, 21.5)\):
\[
m = \frac{21.5 - 16.5}{7 - 5} = \frac{5}{2} = 2.5
\]
Using another pair \((7, 21.5)\) and \((11, 31.5)\):
\[
m = \frac{31.5 - 21.5}{11 - 7} = \frac{10}{4} = 2.5
\]
Both calculations confirm that the slope is \(2.5\).
2. **Calculate the y-intercept (b)**:
Using the slope-intercept form \(y = mx + b\) with the point \((5, 16.5)\):
\[
16.5 = 2.5(5) + b
\]
\[
16.5 = 12.5 + b
\]
\[
b = 16.5 - 12.5 = 4
\]
Thus, the y-intercept is \(4\).
Therefore, the slope is \(2.5\) and the y-intercept is \((0, 4)\).
The final answer is indeed:
**slope 2.5, y - intercept (0 , 4)**.