Question
Using the graph, derive the equation of the line in slope - intercept form
answers:
3x - y = 4
y = -3x - 4
y = 3x - 4
answers:
3x - y = 4
y = -3x - 4
y = 3x - 4
Answers
Answer
you sure
Answered by
GPT-4o mini
To derive the equation of a line in slope-intercept form \(y = mx + b\) using a graph, we need to identify two key components: the slope \(m\) and the y-intercept \(b\).
1. **Identify the Slope (\(m\))**: The slope is determined by the steepness of the line and the direction it goes. If the line rises to the right, the slope is positive. If it falls to the right, the slope is negative. The slope can be calculated as the rise (change in y) over the run (change in x).
2. **Identify the Y-Intercept (\(b\))**: This is the point where the line crosses the y-axis. The y-intercept gives the value of \(y\) when \(x = 0\).
Based on your options:
1. **3x - y = 4**: This is not in slope-intercept form.
2. **y = -3x - 4**: In this case, the slope is \(-3\) and the y-intercept is \(-4\).
3. **y = 3x - 4**: Here, the slope is \(3\) and the y-intercept is \(-4\).
To determine which option is correct, you would need to assess the graph:
- If the line indeed has a steep negative slope and crosses the y-axis at \(-4\), then the equation \(y = -3x - 4\) could be correct.
- If it has a steep positive slope and crosses the y-axis at \(-4\), then the equation \(y = 3x - 4\) would be correct.
Without access to the graph, I cannot definitively determine which of the options is correct, but you can follow this reasoning using the graph's characteristics to select the answer. If you find that the slope is \(3\), the correct answer would be \(y = 3x - 4\). If the slope is \(-3\), it would be \(y = -3x - 4\).
1. **Identify the Slope (\(m\))**: The slope is determined by the steepness of the line and the direction it goes. If the line rises to the right, the slope is positive. If it falls to the right, the slope is negative. The slope can be calculated as the rise (change in y) over the run (change in x).
2. **Identify the Y-Intercept (\(b\))**: This is the point where the line crosses the y-axis. The y-intercept gives the value of \(y\) when \(x = 0\).
Based on your options:
1. **3x - y = 4**: This is not in slope-intercept form.
2. **y = -3x - 4**: In this case, the slope is \(-3\) and the y-intercept is \(-4\).
3. **y = 3x - 4**: Here, the slope is \(3\) and the y-intercept is \(-4\).
To determine which option is correct, you would need to assess the graph:
- If the line indeed has a steep negative slope and crosses the y-axis at \(-4\), then the equation \(y = -3x - 4\) could be correct.
- If it has a steep positive slope and crosses the y-axis at \(-4\), then the equation \(y = 3x - 4\) would be correct.
Without access to the graph, I cannot definitively determine which of the options is correct, but you can follow this reasoning using the graph's characteristics to select the answer. If you find that the slope is \(3\), the correct answer would be \(y = 3x - 4\). If the slope is \(-3\), it would be \(y = -3x - 4\).
Answered by
GPT-4o mini
Let’s clarify the process for finding the equation of a line in slope-intercept form \(y = mx + b\) based on your description of the line's characteristics.
To be more precise in guiding you through this:
1. **Look at the Slope**:
- A **positive slope** means as \(x\) increases, \(y\) increases.
- A **negative slope** means as \(x\) increases, \(y\) decreases.
2. **Identify the Y-Intercept**:
- This is where the line crosses the \(y\)-axis. This value (let's call it \(b\)) is essential for writing the equation.
Now, let's review the potential equations you provided:
1. **\(3x - y = 4\)**:
- This can be rearranged to slope-intercept form: \(y = 3x - 4\). It indicates a slope of \(3\) and a y-intercept of \(-4\).
2. **\(y = -3x - 4\)**:
- Here, the slope is \(-3\) and y-intercept is \(-4\). This indicates the line descends steeply.
3. **\(y = 3x - 4\)**:
- Slope is \(3\) and y-intercept is \(-4\), indicating an ascending line.
For details:
- If the line slopes upwards from left to right, then the equations with positive slopes (\(3x - 4\)) are correct.
- If the line slopes downwards, then the equations with negative slopes (\(-3x - 4\)) are correct.
To confirm which equation is the right one, check the following:
- **Slope direction (positive or negative)**
- **Point where the line crosses the y-axis (the y-intercept)**
If you can provide more context about the graph—like the coordinates of key points or its general behavior (rising or falling)—I can assist you in definitively identifying the correct equation.
To be more precise in guiding you through this:
1. **Look at the Slope**:
- A **positive slope** means as \(x\) increases, \(y\) increases.
- A **negative slope** means as \(x\) increases, \(y\) decreases.
2. **Identify the Y-Intercept**:
- This is where the line crosses the \(y\)-axis. This value (let's call it \(b\)) is essential for writing the equation.
Now, let's review the potential equations you provided:
1. **\(3x - y = 4\)**:
- This can be rearranged to slope-intercept form: \(y = 3x - 4\). It indicates a slope of \(3\) and a y-intercept of \(-4\).
2. **\(y = -3x - 4\)**:
- Here, the slope is \(-3\) and y-intercept is \(-4\). This indicates the line descends steeply.
3. **\(y = 3x - 4\)**:
- Slope is \(3\) and y-intercept is \(-4\), indicating an ascending line.
For details:
- If the line slopes upwards from left to right, then the equations with positive slopes (\(3x - 4\)) are correct.
- If the line slopes downwards, then the equations with negative slopes (\(-3x - 4\)) are correct.
To confirm which equation is the right one, check the following:
- **Slope direction (positive or negative)**
- **Point where the line crosses the y-axis (the y-intercept)**
If you can provide more context about the graph—like the coordinates of key points or its general behavior (rising or falling)—I can assist you in definitively identifying the correct equation.
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