Question
what is (-3,-2) (6,4) as a proper fraction show your work
Answers
GPT-4o mini
To express the points \((-3, -2)\) and \((6, 4)\) as a proper fraction, we first need to understand what this means. In this context, "proper fraction" usually refers to a fraction where the numerator is less than the denominator. Since there are two points given, it looks like you might want to find the slope of the line that connects these two points, which can be expressed as a fraction.
### Steps to find the slope:
1. **Identify the coordinates**:
Point A: \((-3, -2)\)
Point B: \((6, 4)\)
2. **Use the slope formula**:
The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
3. **Plug in the coordinates**:
Set \((x_1, y_1) = (-3, -2)\) and \((x_2, y_2) = (6, 4)\).
So,
\[
m = \frac{4 - (-2)}{6 - (-3)} = \frac{4 + 2}{6 + 3} = \frac{6}{9}
\]
4. **Simplify the fraction**:
The fraction \(\frac{6}{9}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (3):
\[
\frac{6 \div 3}{9 \div 3} = \frac{2}{3}
\]
### Conclusion:
Thus, the slope between the points \((-3, -2)\) and \((6, 4)\) is \(\frac{2}{3}\), which is a proper fraction since the numerator \(2\) is less than the denominator \(3\).
### Steps to find the slope:
1. **Identify the coordinates**:
Point A: \((-3, -2)\)
Point B: \((6, 4)\)
2. **Use the slope formula**:
The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
3. **Plug in the coordinates**:
Set \((x_1, y_1) = (-3, -2)\) and \((x_2, y_2) = (6, 4)\).
So,
\[
m = \frac{4 - (-2)}{6 - (-3)} = \frac{4 + 2}{6 + 3} = \frac{6}{9}
\]
4. **Simplify the fraction**:
The fraction \(\frac{6}{9}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (3):
\[
\frac{6 \div 3}{9 \div 3} = \frac{2}{3}
\]
### Conclusion:
Thus, the slope between the points \((-3, -2)\) and \((6, 4)\) is \(\frac{2}{3}\), which is a proper fraction since the numerator \(2\) is less than the denominator \(3\).
now words just the work
GPT-4o mini
To find the slope between the points (-3, -2) and (6, 4), we first identify the coordinates of the two points. The first point is called A with coordinates (-3, -2) and the second point is called B with coordinates (6, 4).
Next, we use the slope formula, which is defined as the difference in the y-coordinates divided by the difference in the x-coordinates. In this case, we subtract the y-coordinate of point A from the y-coordinate of point B and then subtract the x-coordinate of point A from the x-coordinate of point B.
By substituting the coordinates into the formula, we calculate the difference in the y-coordinates as 4 minus (-2), which simplifies to 4 + 2, giving us 6. For the difference in the x-coordinates, we compute 6 minus (-3), which simplifies to 6 + 3, resulting in 9.
Now we have the slope represented as the fraction 6 over 9. The next step is to simplify this fraction by finding the greatest common divisor of the numerator and the denominator, which is 3. Dividing both the numerator and the denominator by 3 gives us the simplified fraction of 2 over 3.
In conclusion, the slope between the points (-3, -2) and (6, 4) is expressed as the proper fraction 2/3, as the numerator (2) is less than the denominator (3).
Next, we use the slope formula, which is defined as the difference in the y-coordinates divided by the difference in the x-coordinates. In this case, we subtract the y-coordinate of point A from the y-coordinate of point B and then subtract the x-coordinate of point A from the x-coordinate of point B.
By substituting the coordinates into the formula, we calculate the difference in the y-coordinates as 4 minus (-2), which simplifies to 4 + 2, giving us 6. For the difference in the x-coordinates, we compute 6 minus (-3), which simplifies to 6 + 3, resulting in 9.
Now we have the slope represented as the fraction 6 over 9. The next step is to simplify this fraction by finding the greatest common divisor of the numerator and the denominator, which is 3. Dividing both the numerator and the denominator by 3 gives us the simplified fraction of 2 over 3.
In conclusion, the slope between the points (-3, -2) and (6, 4) is expressed as the proper fraction 2/3, as the numerator (2) is less than the denominator (3).
just the numbers
GPT-4o mini
1. Points: (-3, -2) and (6, 4)
2. Slope formula: \(\frac{y_2 - y_1}{x_2 - x_1}\)
3. Substituting: \(\frac{4 - (-2)}{6 - (-3)}\)
4. Simplifying: \(\frac{4 + 2}{6 + 3} = \frac{6}{9}\)
5. Simplifying further: \(\frac{6 \div 3}{9 \div 3} = \frac{2}{3}\)
6. Result: \(\frac{2}{3}\)
2. Slope formula: \(\frac{y_2 - y_1}{x_2 - x_1}\)
3. Substituting: \(\frac{4 - (-2)}{6 - (-3)}\)
4. Simplifying: \(\frac{4 + 2}{6 + 3} = \frac{6}{9}\)
5. Simplifying further: \(\frac{6 \div 3}{9 \div 3} = \frac{2}{3}\)
6. Result: \(\frac{2}{3}\)
(-2,1) (2,7) as a proper fraction show your work just the numbers
GPT-4o mini
1. Points: (-2, 1) and (2, 7)
2. Slope formula: \(\frac{y_2 - y_1}{x_2 - x_1}\)
3. Substituting: \(\frac{7 - 1}{2 - (-2)}\)
4. Simplifying: \(\frac{6}{2 + 2} = \frac{6}{4}\)
5. Simplifying further: \(\frac{6 \div 2}{4 \div 2} = \frac{3}{2}\)
6. Result: \(\frac{3}{2}\)
2. Slope formula: \(\frac{y_2 - y_1}{x_2 - x_1}\)
3. Substituting: \(\frac{7 - 1}{2 - (-2)}\)
4. Simplifying: \(\frac{6}{2 + 2} = \frac{6}{4}\)
5. Simplifying further: \(\frac{6 \div 2}{4 \div 2} = \frac{3}{2}\)
6. Result: \(\frac{3}{2}\)
(-5,-2) (0,1) as a proper fraction
GPT-4o mini
1. Points: (-5, -2) and (0, 1)
2. Slope formula: \(\frac{y_2 - y_1}{x_2 - x_1}\)
3. Substituting: \(\frac{1 - (-2)}{0 - (-5)}\)
4. Simplifying: \(\frac{1 + 2}{0 + 5} = \frac{3}{5}\)
5. Result: \(\frac{3}{5}\)
2. Slope formula: \(\frac{y_2 - y_1}{x_2 - x_1}\)
3. Substituting: \(\frac{1 - (-2)}{0 - (-5)}\)
4. Simplifying: \(\frac{1 + 2}{0 + 5} = \frac{3}{5}\)
5. Result: \(\frac{3}{5}\)
(-1,-3) 2,6 as a proper fraction
GPT-4o mini
1. Points: \((-1, -3)\) and \((2, 6)\)
2. Slope formula: \(\frac{y_2 - y_1}{x_2 - x_1}\)
3. Substituting: \(\frac{6 - (-3)}{2 - (-1)}\)
4. Simplifying: \(\frac{6 + 3}{2 + 1} = \frac{9}{3}\)
5. Simplifying further: \(\frac{9 \div 3}{3 \div 3} = \frac{3}{1}\)
6. Result: \(\frac{3}{1}\) (which is not a proper fraction, but an improper fraction)
2. Slope formula: \(\frac{y_2 - y_1}{x_2 - x_1}\)
3. Substituting: \(\frac{6 - (-3)}{2 - (-1)}\)
4. Simplifying: \(\frac{6 + 3}{2 + 1} = \frac{9}{3}\)
5. Simplifying further: \(\frac{9 \div 3}{3 \div 3} = \frac{3}{1}\)
6. Result: \(\frac{3}{1}\) (which is not a proper fraction, but an improper fraction)