Question
Find the slope of the line that passes through the pair of points (1,7), (10, 1)
Answers
Answer
So the snare is (2,3)?
Answer
Oh ok so the line that passes through the points (1, 7) and (10, 1) would be (2,3) is that right?
Answer
Oh ok I don’t think I’m sure then, could you explain it again in simple terms please thank you
Answered by
GPT-4o mini
To find the slope of the line that passes through the points \((1, 7)\) and \((10, 1)\), we use the formula for the slope \(m\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Where \((x_1, y_1) = (1, 7)\) and \((x_2, y_2) = (10, 1)\).
Substituting in the values:
\[
m = \frac{1 - 7}{10 - 1}
\]
Calculating the numerator and the denominator:
\[
m = \frac{-6}{9}
\]
Simplifying the fraction:
\[
m = -\frac{2}{3}
\]
Thus, the slope of the line that passes through the points \((1, 7)\) and \((10, 1)\) is
\[
\boxed{-\frac{2}{3}}.
\]
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Where \((x_1, y_1) = (1, 7)\) and \((x_2, y_2) = (10, 1)\).
Substituting in the values:
\[
m = \frac{1 - 7}{10 - 1}
\]
Calculating the numerator and the denominator:
\[
m = \frac{-6}{9}
\]
Simplifying the fraction:
\[
m = -\frac{2}{3}
\]
Thus, the slope of the line that passes through the points \((1, 7)\) and \((10, 1)\) is
\[
\boxed{-\frac{2}{3}}.
\]
Answered by
GPT-4o mini
It seems there may have been a bit of confusion. The slope we calculated is \(-\frac{2}{3}\), which describes the steepness of the line passing through the points (1, 7) and (10, 1).
If you're referring to the coordinates of a point, (2, 3) is not directly related to the slope we calculated. The slope itself is a single numerical value indicating how steep the line is.
If you need further clarification or if you have a different question regarding points or slopes, please let me know!
If you're referring to the coordinates of a point, (2, 3) is not directly related to the slope we calculated. The slope itself is a single numerical value indicating how steep the line is.
If you need further clarification or if you have a different question regarding points or slopes, please let me know!
Answered by
GPT-4o mini
The point (2, 3) is not on the line that passes through the points (1, 7) and (10, 1). To clarify, the line defined by the two given points does not necessarily pass through (2, 3).
To check if (2, 3) lies on the line defined by the two points, we can find the equation of the line and see if (2, 3) satisfies it.
We have the slope \( m = -\frac{2}{3} \). Using point-slope form, we can write the equation of the line in point-slope form using one of our points, say (1, 7):
\[
y - 7 = -\frac{2}{3}(x - 1)
\]
Expanding this gives:
\[
y - 7 = -\frac{2}{3}x + \frac{2}{3}
\]
\[
y = -\frac{2}{3}x + \frac{2}{3} + 7
\]
\[
y = -\frac{2}{3}x + \frac{2}{3} + \frac{21}{3}
\]
\[
y = -\frac{2}{3}x + \frac{23}{3}
\]
Now, we can substitute \( x = 2 \) into this equation to see if \( y \) equals 3:
\[
y = -\frac{2}{3}(2) + \frac{23}{3}
\]
\[
y = -\frac{4}{3} + \frac{23}{3}
\]
\[
y = \frac{19}{3}
\]
Since \( \frac{19}{3} \) does not equal 3, the point (2, 3) does not lie on the line defined by the points (1, 7) and (10, 1).
Thus, (2, 3) is not a point on the line. If you need any further assistance or clarification, feel free to ask!
To check if (2, 3) lies on the line defined by the two points, we can find the equation of the line and see if (2, 3) satisfies it.
We have the slope \( m = -\frac{2}{3} \). Using point-slope form, we can write the equation of the line in point-slope form using one of our points, say (1, 7):
\[
y - 7 = -\frac{2}{3}(x - 1)
\]
Expanding this gives:
\[
y - 7 = -\frac{2}{3}x + \frac{2}{3}
\]
\[
y = -\frac{2}{3}x + \frac{2}{3} + 7
\]
\[
y = -\frac{2}{3}x + \frac{2}{3} + \frac{21}{3}
\]
\[
y = -\frac{2}{3}x + \frac{23}{3}
\]
Now, we can substitute \( x = 2 \) into this equation to see if \( y \) equals 3:
\[
y = -\frac{2}{3}(2) + \frac{23}{3}
\]
\[
y = -\frac{4}{3} + \frac{23}{3}
\]
\[
y = \frac{19}{3}
\]
Since \( \frac{19}{3} \) does not equal 3, the point (2, 3) does not lie on the line defined by the points (1, 7) and (10, 1).
Thus, (2, 3) is not a point on the line. If you need any further assistance or clarification, feel free to ask!
Answered by
GPT-4o mini
Of course! Let’s break it down step by step in simple terms:
1. **Understanding Points**: You have two points: (1, 7) and (10, 1). These points have coordinates that tell us their position on a graph:
- The first number (x-coordinate) tells us how far to go left or right.
- The second number (y-coordinate) tells us how far to go up or down.
2. **Finding the Slope**: The slope of a line tells us how steep it is. We calculated it using these two points:
- Slope formula:
\[
m = \frac{(y_2 - y_1)}{(x_2 - x_1)}
\]
- For your points, we used:
- \((x_1, y_1) = (1, 7)\)
- \((x_2, y_2) = (10, 1)\)
- Plugging them into the formula:
\[
m = \frac{(1 - 7)}{(10 - 1)} = \frac{-6}{9} = -\frac{2}{3}
\]
- This means the line slopes downward, going down 2 units for every 3 units it goes to the right.
3. **Finding the Line’s Equation**: With the slope calculated, we can find the equation of the line:
- We use one of the points (let’s use (1, 7)) and the slope \(-\frac{2}{3}\):
- The equation in point-slope form is:
\[
y - 7 = -\frac{2}{3}(x - 1)
\]
- This can be rearranged to find a general equation that describes all points on the line.
4. **Checking the Point (2, 3)**: Now, let’s see if the point (2, 3) lies on this line.
- To do this, we substitute \( x = 2 \) into the line’s equation and see what \( y \) equals:
\[
y = -\frac{2}{3}(2) + \frac{23}{3}
\]
- This calculation gives us \( y = \frac{19}{3} \), which is not equal to 3. Therefore, (2, 3) is not on the line formed by (1, 7) and (10, 1).
### Summary
- The slope found was \(-\frac{2}{3}\).
- The line does **not** contain the point (2, 3).
If you have any more questions or need further clarification, feel free to ask!
1. **Understanding Points**: You have two points: (1, 7) and (10, 1). These points have coordinates that tell us their position on a graph:
- The first number (x-coordinate) tells us how far to go left or right.
- The second number (y-coordinate) tells us how far to go up or down.
2. **Finding the Slope**: The slope of a line tells us how steep it is. We calculated it using these two points:
- Slope formula:
\[
m = \frac{(y_2 - y_1)}{(x_2 - x_1)}
\]
- For your points, we used:
- \((x_1, y_1) = (1, 7)\)
- \((x_2, y_2) = (10, 1)\)
- Plugging them into the formula:
\[
m = \frac{(1 - 7)}{(10 - 1)} = \frac{-6}{9} = -\frac{2}{3}
\]
- This means the line slopes downward, going down 2 units for every 3 units it goes to the right.
3. **Finding the Line’s Equation**: With the slope calculated, we can find the equation of the line:
- We use one of the points (let’s use (1, 7)) and the slope \(-\frac{2}{3}\):
- The equation in point-slope form is:
\[
y - 7 = -\frac{2}{3}(x - 1)
\]
- This can be rearranged to find a general equation that describes all points on the line.
4. **Checking the Point (2, 3)**: Now, let’s see if the point (2, 3) lies on this line.
- To do this, we substitute \( x = 2 \) into the line’s equation and see what \( y \) equals:
\[
y = -\frac{2}{3}(2) + \frac{23}{3}
\]
- This calculation gives us \( y = \frac{19}{3} \), which is not equal to 3. Therefore, (2, 3) is not on the line formed by (1, 7) and (10, 1).
### Summary
- The slope found was \(-\frac{2}{3}\).
- The line does **not** contain the point (2, 3).
If you have any more questions or need further clarification, feel free to ask!
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