Question
What is the slope of the line that passes through the points left parenthesis, 4, comma, 1, right parenthesis(4,1) and left parenthesis, 0, comma, minus, 19, right parenthesis(0,−19)? Write your answer in simplest form
Answers
GPT-4o mini
To find the slope of the line that passes through the points \((4, 1)\) and \((0, -19)\), we can use the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, we can assign the points as follows:
- \((x_1, y_1) = (4, 1)\)
- \((x_2, y_2) = (0, -19)\)
Now substitute the coordinates into the formula:
\[
m = \frac{-19 - 1}{0 - 4}
\]
Calculating the numerator and the denominator:
\[
m = \frac{-20}{-4}
\]
Now simplify:
\[
m = \frac{-20 \div -4}{-4 \div -4} = \frac{5}{1} = 5
\]
Thus, the slope of the line is
\[
\boxed{5}
\]
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, we can assign the points as follows:
- \((x_1, y_1) = (4, 1)\)
- \((x_2, y_2) = (0, -19)\)
Now substitute the coordinates into the formula:
\[
m = \frac{-19 - 1}{0 - 4}
\]
Calculating the numerator and the denominator:
\[
m = \frac{-20}{-4}
\]
Now simplify:
\[
m = \frac{-20 \div -4}{-4 \div -4} = \frac{5}{1} = 5
\]
Thus, the slope of the line is
\[
\boxed{5}
\]
Graph the following features:
Slope = one third
3
1
Y-intercept = minus, 5−5
Slope = one third
3
1
Y-intercept = minus, 5−5
GPT-4o mini
To graph a line with a slope of \(\frac{1}{3}\) and a y-intercept of \(-5\), you can follow these steps:
1. **Identify the y-intercept**: The y-intercept is the point where the line crosses the y-axis. In this case, the y-intercept is \(-5\), which corresponds to the point \((0, -5)\). Plot this point on the graph.
2. **Use the slope**: The slope is \(\frac{1}{3}\), which means that for every 1 unit you move up (rise), you move 3 units to the right (run). From the y-intercept point \((0, -5)\):
- Move up 1 unit to \(y = -4\) (i.e., from \(-5\) to \(-4\)).
- Move 3 units to the right to \(x = 3\) (i.e., from \(0\) to \(3\)).
This gives you the point \((3, -4)\). Plot this point.
3. **Draw the line**: Draw a straight line through the points \((0, -5)\) and \((3, -4)\). Extend the line in both directions.
Here's a summary of the key points to plot:
- **Y-intercept**: \((0, -5)\)
- **Next point based on slope**: \((3, -4)\)
### Visualization
While I cannot provide a visual graph directly, you can sketch it as follows:
1. Mark the point \((0, -5)\) on the y-axis.
2. Move to the right and slightly upwards to locate the point \((3, -4)\).
3. Draw a line through these two points, extending it in both directions.
This should give you a line with a slope of \(\frac{1}{3}\) that crosses the y-axis at \(-5\).
1. **Identify the y-intercept**: The y-intercept is the point where the line crosses the y-axis. In this case, the y-intercept is \(-5\), which corresponds to the point \((0, -5)\). Plot this point on the graph.
2. **Use the slope**: The slope is \(\frac{1}{3}\), which means that for every 1 unit you move up (rise), you move 3 units to the right (run). From the y-intercept point \((0, -5)\):
- Move up 1 unit to \(y = -4\) (i.e., from \(-5\) to \(-4\)).
- Move 3 units to the right to \(x = 3\) (i.e., from \(0\) to \(3\)).
This gives you the point \((3, -4)\). Plot this point.
3. **Draw the line**: Draw a straight line through the points \((0, -5)\) and \((3, -4)\). Extend the line in both directions.
Here's a summary of the key points to plot:
- **Y-intercept**: \((0, -5)\)
- **Next point based on slope**: \((3, -4)\)
### Visualization
While I cannot provide a visual graph directly, you can sketch it as follows:
1. Mark the point \((0, -5)\) on the y-axis.
2. Move to the right and slightly upwards to locate the point \((3, -4)\).
3. Draw a line through these two points, extending it in both directions.
This should give you a line with a slope of \(\frac{1}{3}\) that crosses the y-axis at \(-5\).
Graph the line with the equation y, equals, one half, x, plus, 5y=
2
1
x+5.
2
1
x+5.
GPT-4o mini
To graph the line with the equation
\[
y = \frac{1}{2}x + 5,
\]
follow these steps:
### Step 1: Identify the y-intercept
The y-intercept is the point where the line crosses the y-axis. In the equation \(y = \frac{1}{2}x + 5\), the y-intercept is \(5\), which corresponds to the point \((0, 5)\). Plot this point on the graph.
### Step 2: Identify the slope
The slope of the line is \(\frac{1}{2}\). This means that for every 1 unit you move up (rise), you move 2 units to the right (run).
### Step 3: Plot another point using the slope
From the y-intercept \((0, 5)\), apply the slope:
- Move up 1 unit to \(y = 6\).
- Move 2 units to the right to \(x = 2\).
Now, you have another point: \((2, 6)\). Plot this point as well.
### Step 4: Draw the line
Now that you have two points \((0, 5)\) and \((2, 6)\), draw a straight line through these points, extending it in both directions.
### Summary of key points
- **Y-intercept**: \((0, 5)\)
- **Second point from the slope**: \((2, 6)\)
### Visualization
While I can't create visual graphics directly, you can sketch the line by following these instructions on graph paper or using a graphing tool. The line will rise gradually from left to right due to the positive slope of \(\frac{1}{2}\).
\[
y = \frac{1}{2}x + 5,
\]
follow these steps:
### Step 1: Identify the y-intercept
The y-intercept is the point where the line crosses the y-axis. In the equation \(y = \frac{1}{2}x + 5\), the y-intercept is \(5\), which corresponds to the point \((0, 5)\). Plot this point on the graph.
### Step 2: Identify the slope
The slope of the line is \(\frac{1}{2}\). This means that for every 1 unit you move up (rise), you move 2 units to the right (run).
### Step 3: Plot another point using the slope
From the y-intercept \((0, 5)\), apply the slope:
- Move up 1 unit to \(y = 6\).
- Move 2 units to the right to \(x = 2\).
Now, you have another point: \((2, 6)\). Plot this point as well.
### Step 4: Draw the line
Now that you have two points \((0, 5)\) and \((2, 6)\), draw a straight line through these points, extending it in both directions.
### Summary of key points
- **Y-intercept**: \((0, 5)\)
- **Second point from the slope**: \((2, 6)\)
### Visualization
While I can't create visual graphics directly, you can sketch the line by following these instructions on graph paper or using a graphing tool. The line will rise gradually from left to right due to the positive slope of \(\frac{1}{2}\).
Write the equation of the line in fully simplified slope-intercept form.
-8 was the Y intercept (-3,2) (2,4)
-8 was the Y intercept (-3,2) (2,4)
GPT-4o mini
To find the equation of the line in slope-intercept form \(y = mx + b\) using the given information, we will follow these steps:
1. **Identify the slope (\(m\))** using the points \((-3, 2)\) and \((2, 4)\).
2. **Use the y-intercept**, which is given as \(-8\).
3. **Write the equation** in slope-intercept form.
### Step 1: Calculate the slope (\(m\))
The slope \(m\) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-3, 2)\) and \((2, 4)\):
- Let \((x_1, y_1) = (-3, 2)\)
- Let \((x_2, y_2) = (2, 4)\)
Now substitute the values into the slope formula:
\[
m = \frac{4 - 2}{2 - (-3)} = \frac{2}{2 + 3} = \frac{2}{5}
\]
### Step 2: Use the y-intercept
Given the y-intercept \(b = -8\).
### Step 3: Write the equation in slope-intercept form
Now we can write the equation using the slope \(m = \frac{2}{5}\) and the y-intercept \(b = -8\):
\[
y = \frac{2}{5}x - 8
\]
### Final Equation
The equation of the line in fully simplified slope-intercept form is
\[
\boxed{y = \frac{2}{5}x - 8}
\]
1. **Identify the slope (\(m\))** using the points \((-3, 2)\) and \((2, 4)\).
2. **Use the y-intercept**, which is given as \(-8\).
3. **Write the equation** in slope-intercept form.
### Step 1: Calculate the slope (\(m\))
The slope \(m\) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-3, 2)\) and \((2, 4)\):
- Let \((x_1, y_1) = (-3, 2)\)
- Let \((x_2, y_2) = (2, 4)\)
Now substitute the values into the slope formula:
\[
m = \frac{4 - 2}{2 - (-3)} = \frac{2}{2 + 3} = \frac{2}{5}
\]
### Step 2: Use the y-intercept
Given the y-intercept \(b = -8\).
### Step 3: Write the equation in slope-intercept form
Now we can write the equation using the slope \(m = \frac{2}{5}\) and the y-intercept \(b = -8\):
\[
y = \frac{2}{5}x - 8
\]
### Final Equation
The equation of the line in fully simplified slope-intercept form is
\[
\boxed{y = \frac{2}{5}x - 8}
\]
incorrect
GPT-4o mini
I apologize for the misunderstanding. Let's revisit the problem using the provided information correctly.
### Given Information:
- Points: \((-3, 2)\) and \((2, 4)\)
- Y-intercept: \(-8\)
### Step 1: Calculate the slope (\(m\))
Using the points \((-3, 2)\) and \((2, 4)\):
- Let \((x_1, y_1) = (-3, 2)\)
- Let \((x_2, y_2) = (2, 4)\)
Now substitute into the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{2 - (-3)} = \frac{2}{2 + 3} = \frac{2}{5}
\]
Correctly, the slope \(m = \frac{2}{5}\).
### Step 2: Use the given y-intercept
The y-intercept is given as \(-8\).
### Step 3: Write the equation in slope-intercept form
Now we can express the equation using the slope and y-intercept:
\[
y = mx + b
\]
Substituting \(m = \frac{2}{5}\) and \(b = -8\):
\[
y = \frac{2}{5}x - 8
\]
### Confirming Information
It's essential to note that even though the slope calculation was right, the y-intercept of \(-8\) contradicts the \(y\)-values of the points provided, which would suggest the line should be through those points. This means the confirmation doesn’t seem correct since a line passing through both \((-3, 2)\) and \((2, 4)\) should have its own y-intercept, not the one provided as \(-8\).
If both points are to be connected with a line with a y-intercept of \(-8\), it would suggest a different approach to derive an accurate intersection, not just from those coordinates.
### Conclusion
Given both original points, here's the simpler form reinforced:
The **correct equation** based on the slope calculated between the given points is:
\[
\boxed{y = \frac{2}{5}x + \frac{8}{5}}
\]
If your condition was sending the line through that specific y-intercept, then we'd need more information or context to assist further. If there's an oversight in what you're requesting, feel free to clarify so we can get this right!
### Given Information:
- Points: \((-3, 2)\) and \((2, 4)\)
- Y-intercept: \(-8\)
### Step 1: Calculate the slope (\(m\))
Using the points \((-3, 2)\) and \((2, 4)\):
- Let \((x_1, y_1) = (-3, 2)\)
- Let \((x_2, y_2) = (2, 4)\)
Now substitute into the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{2 - (-3)} = \frac{2}{2 + 3} = \frac{2}{5}
\]
Correctly, the slope \(m = \frac{2}{5}\).
### Step 2: Use the given y-intercept
The y-intercept is given as \(-8\).
### Step 3: Write the equation in slope-intercept form
Now we can express the equation using the slope and y-intercept:
\[
y = mx + b
\]
Substituting \(m = \frac{2}{5}\) and \(b = -8\):
\[
y = \frac{2}{5}x - 8
\]
### Confirming Information
It's essential to note that even though the slope calculation was right, the y-intercept of \(-8\) contradicts the \(y\)-values of the points provided, which would suggest the line should be through those points. This means the confirmation doesn’t seem correct since a line passing through both \((-3, 2)\) and \((2, 4)\) should have its own y-intercept, not the one provided as \(-8\).
If both points are to be connected with a line with a y-intercept of \(-8\), it would suggest a different approach to derive an accurate intersection, not just from those coordinates.
### Conclusion
Given both original points, here's the simpler form reinforced:
The **correct equation** based on the slope calculated between the given points is:
\[
\boxed{y = \frac{2}{5}x + \frac{8}{5}}
\]
If your condition was sending the line through that specific y-intercept, then we'd need more information or context to assist further. If there's an oversight in what you're requesting, feel free to clarify so we can get this right!