Question
The table shows the linear relationship between the balance saved each week. What is the rate of change in terms of balance (in dollars) with respect to the number of weeks?
week 0,1, 3, 6, 8, 13
Balance 32, 39, 53, 74, 88, 123
week 0,1, 3, 6, 8, 13
Balance 32, 39, 53, 74, 88, 123
Answers
Answer
The graph of a line is represented by the equation `5x-8y=40.`What value represents the rate of change of `y` with respect to `x` for the equation?
Answer
Identify the slope and y intercept. Then, graph the linear equation.
`y=-\frac{5}{3}x+4`
Begin with `b,`the y-intercept:
Move with `m,`the slope:
`y=-\frac{5}{3}x+4`
Begin with `b,`the y-intercept:
Move with `m,`the slope:
Answer
how would I label the graph
Answer
Identify the slope and y intercept. Then, graph the linear equation.
`y-5=-3x`
Begin with `b,`the y-intercept:
Move with `m,`the slope:
`y-5=-3x`
Begin with `b,`the y-intercept:
Move with `m,`the slope:
Answer
how do I label the graph
Answered by
GPT-4o mini
To find the rate of change in terms of balance (in dollars) with respect to the number of weeks, we can calculate the slope of the linear relationship between the number of weeks and the balance saved.
The slope (rate of change) can be calculated using the formula:
\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \( y \) represents the balance and \( x \) represents the number of weeks.
Using two points from the provided data, we can choose:
1. Week 0, Balance 32
2. Week 13, Balance 123
Now we plug these values into the slope formula:
\[
\text{slope} = \frac{123 - 32}{13 - 0} = \frac{91}{13} \approx 7
\]
Therefore, the rate of change in terms of balance with respect to the number of weeks is approximately **7 dollars per week**.
The slope (rate of change) can be calculated using the formula:
\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \( y \) represents the balance and \( x \) represents the number of weeks.
Using two points from the provided data, we can choose:
1. Week 0, Balance 32
2. Week 13, Balance 123
Now we plug these values into the slope formula:
\[
\text{slope} = \frac{123 - 32}{13 - 0} = \frac{91}{13} \approx 7
\]
Therefore, the rate of change in terms of balance with respect to the number of weeks is approximately **7 dollars per week**.
Answered by
GPT-4o mini
To find the rate of change of \( y \) with respect to \( x \) for the equation \( 5x - 8y = 40 \), we need to rewrite the equation in slope-intercept form, which is \( y = mx + b \), where \( m \) represents the slope (the rate of change of \( y \) with respect to \( x \)).
Starting with the original equation:
\[
5x - 8y = 40
\]
We want to isolate \( y \). First, we can rearrange the equation:
\[
-8y = -5x + 40
\]
Now, divide every term by -8:
\[
y = \frac{5}{8}x - 5
\]
Now the equation is in slope-intercept form. The slope \( m \) is \( \frac{5}{8} \).
Therefore, the rate of change of \( y \) with respect to \( x \) is:
\[
\frac{5}{8}
\]
Starting with the original equation:
\[
5x - 8y = 40
\]
We want to isolate \( y \). First, we can rearrange the equation:
\[
-8y = -5x + 40
\]
Now, divide every term by -8:
\[
y = \frac{5}{8}x - 5
\]
Now the equation is in slope-intercept form. The slope \( m \) is \( \frac{5}{8} \).
Therefore, the rate of change of \( y \) with respect to \( x \) is:
\[
\frac{5}{8}
\]
Answered by
GPT-4o mini
To identify the slope and y-intercept of the linear equation \( y = -\frac{5}{3}x + 4 \), we can analyze the equation in slope-intercept form, \( y = mx + b \), where:
- \( m \) is the slope
- \( b \) is the y-intercept
### Identification:
- **Y-Intercept (b)**: The y-intercept is \( 4 \). This means the line crosses the y-axis at the point \( (0, 4) \).
- **Slope (m)**: The slope is \( -\frac{5}{3} \). This is the rate at which \( y \) changes for a change in \( x \). Specifically, for every increase of \( 3 \) in \( x \), \( y \) decreases by \( 5 \).
### Graphing the Equation:
1. **Plot the Y-Intercept**:
- Start at the point \( (0, 4) \) on the graph.
2. **Use the Slope**:
- From the y-intercept \( (0, 4) \), use the slope \(-\frac{5}{3}\):
- Move down \( 5 \) units (because the slope is negative).
- Move right \( 3 \) units.
- This brings you to the point \( (3, -1) \).
3. **Draw the Line**:
- After plotting the points \( (0, 4) \) and \( (3, -1) \), draw a straight line through these points.
Here’s a summary of the points to plot:
- Y-Intercept: \( (0, 4) \)
- Second point using the slope: \( (3, -1) \)
### Additional Points (Optional):
To provide additional accuracy, you can find more points by continuing to use the slope:
- From \( (3, -1) \):
- Move down \( 5 \) units and right \( 3 \) units:
- This brings you to \( (6, -6) \).
You can also move in the opposite direction:
- From \( (0, 4) \):
- Move up \( 5 \) units and left \( 3 \) units:
- This takes you to \( (-3, 9) \).
### Summary:
- **Slope (m)**: \(-\frac{5}{3}\)
- **Y-Intercept (b)**: \(4\)
The graphic representation should show a straight line descending from left to right, crossing the y-axis at \( (0, 4) \) and passing through the other points you plotted.
- \( m \) is the slope
- \( b \) is the y-intercept
### Identification:
- **Y-Intercept (b)**: The y-intercept is \( 4 \). This means the line crosses the y-axis at the point \( (0, 4) \).
- **Slope (m)**: The slope is \( -\frac{5}{3} \). This is the rate at which \( y \) changes for a change in \( x \). Specifically, for every increase of \( 3 \) in \( x \), \( y \) decreases by \( 5 \).
### Graphing the Equation:
1. **Plot the Y-Intercept**:
- Start at the point \( (0, 4) \) on the graph.
2. **Use the Slope**:
- From the y-intercept \( (0, 4) \), use the slope \(-\frac{5}{3}\):
- Move down \( 5 \) units (because the slope is negative).
- Move right \( 3 \) units.
- This brings you to the point \( (3, -1) \).
3. **Draw the Line**:
- After plotting the points \( (0, 4) \) and \( (3, -1) \), draw a straight line through these points.
Here’s a summary of the points to plot:
- Y-Intercept: \( (0, 4) \)
- Second point using the slope: \( (3, -1) \)
### Additional Points (Optional):
To provide additional accuracy, you can find more points by continuing to use the slope:
- From \( (3, -1) \):
- Move down \( 5 \) units and right \( 3 \) units:
- This brings you to \( (6, -6) \).
You can also move in the opposite direction:
- From \( (0, 4) \):
- Move up \( 5 \) units and left \( 3 \) units:
- This takes you to \( (-3, 9) \).
### Summary:
- **Slope (m)**: \(-\frac{5}{3}\)
- **Y-Intercept (b)**: \(4\)
The graphic representation should show a straight line descending from left to right, crossing the y-axis at \( (0, 4) \) and passing through the other points you plotted.
Answered by
GPT-4o mini
Labeling a graph effectively involves clearly marking important features including the axes, the title, the points, and any lines or slopes. Here’s a step-by-step guide to labeling the graph of the equation \( y = -\frac{5}{3}x + 4 \):
### Steps to Label the Graph
1. **Title the Graph**:
- Write a title at the top of the graph that describes what the graph represents. For example, you could write:
- **"Graph of the Linear Equation \( y = -\frac{5}{3}x + 4 \)"**
2. **Label the Axes**:
- The horizontal axis (x-axis) should be labeled as:
- **"x"**
- The vertical axis (y-axis) should be labeled as:
- **"y"**
3. **Plot Points**:
- Clearly mark the points you plotted:
- \( (0, 4) \) - This is the y-intercept. You can label this point as:
- **"Y-Intercept (0, 4)"**
- \( (3, -1) \) - You could label this point as:
- **"Point (3, -1)"**
- \( (6, -6) \) and any additional points you plotted can be labeled similarly.
4. **Draw the Line**:
- Draw a straight line through the points you plotted, extending the line to ensure it's clear it continues beyond just the plotted points.
5. **Slope Indicator (Optional)**:
- You can indicate the slope on the graph if you want to emphasize how the slope affects the direction of the line:
- Draw a small arrow along the rise and run corresponding to the slope. For example, from \( (0, 4) \) going down \( 5 \) units (to \( (3, -1) \)) and right \( 3 \) units. You could label it:
- **"Slope: -\(\frac{5}{3}\)"**
6. **Grid (If applicable)**:
- If you're working on graph paper or using graphing software, make sure the grid is clear as it helps to read values more easily. Each intersection of the grid lines can correspond to the values of \( x \) and \( y \).
7. **Ensure Clarity**:
- Make sure all labels are clearly written and not too close to the points so they remain legible. Use a consistent font size and style.
### Example Summary of Labels:
- **Title**: "Graph of the Linear Equation \( y = -\frac{5}{3}x + 4 \)"
- **X-Axis Label**: "x"
- **Y-Axis Label**: "y"
- **Point Labels**: "Y-Intercept (0, 4)", "Point (3, -1)", "Point (6, -6)", etc.
- **Slope Indicator**: "Slope: -\(\frac{5}{3}\)"
By following these steps, you will create a well-labeled graph that effectively communicates the key information about the linear equation you're studying.
### Steps to Label the Graph
1. **Title the Graph**:
- Write a title at the top of the graph that describes what the graph represents. For example, you could write:
- **"Graph of the Linear Equation \( y = -\frac{5}{3}x + 4 \)"**
2. **Label the Axes**:
- The horizontal axis (x-axis) should be labeled as:
- **"x"**
- The vertical axis (y-axis) should be labeled as:
- **"y"**
3. **Plot Points**:
- Clearly mark the points you plotted:
- \( (0, 4) \) - This is the y-intercept. You can label this point as:
- **"Y-Intercept (0, 4)"**
- \( (3, -1) \) - You could label this point as:
- **"Point (3, -1)"**
- \( (6, -6) \) and any additional points you plotted can be labeled similarly.
4. **Draw the Line**:
- Draw a straight line through the points you plotted, extending the line to ensure it's clear it continues beyond just the plotted points.
5. **Slope Indicator (Optional)**:
- You can indicate the slope on the graph if you want to emphasize how the slope affects the direction of the line:
- Draw a small arrow along the rise and run corresponding to the slope. For example, from \( (0, 4) \) going down \( 5 \) units (to \( (3, -1) \)) and right \( 3 \) units. You could label it:
- **"Slope: -\(\frac{5}{3}\)"**
6. **Grid (If applicable)**:
- If you're working on graph paper or using graphing software, make sure the grid is clear as it helps to read values more easily. Each intersection of the grid lines can correspond to the values of \( x \) and \( y \).
7. **Ensure Clarity**:
- Make sure all labels are clearly written and not too close to the points so they remain legible. Use a consistent font size and style.
### Example Summary of Labels:
- **Title**: "Graph of the Linear Equation \( y = -\frac{5}{3}x + 4 \)"
- **X-Axis Label**: "x"
- **Y-Axis Label**: "y"
- **Point Labels**: "Y-Intercept (0, 4)", "Point (3, -1)", "Point (6, -6)", etc.
- **Slope Indicator**: "Slope: -\(\frac{5}{3}\)"
By following these steps, you will create a well-labeled graph that effectively communicates the key information about the linear equation you're studying.
Answered by
GPT-4o mini
To analyze the equation \( y - 5 = -3x \), we will first identify the slope and y-intercept, and then provide steps for graphing the equation.
### Step 1: Identify the Slope and Y-Intercept
#### Rewrite in Slope-Intercept Form
The equation given is in a form that resembles point-slope form. To put it in slope-intercept form (\( y = mx + b \)), where \( m \) is the slope and \( b \) is the y-intercept, we can rearrange it:
\[
y - 5 = -3x
\]
Adding \( 5 \) to both sides:
\[
y = -3x + 5
\]
### Identification:
- **Y-Intercept (b)**: The y-intercept is \( 5 \). This means the line crosses the y-axis at the point \( (0, 5) \).
- **Slope (m)**: The slope is \( -3 \). This indicates that for every increase of \( 1 \) in \( x \), \( y \) decreases by \( 3 \) (a downward slope).
### Step 2: Graphing the Equation
1. **Plot the Y-Intercept**:
- Start at the point \( (0, 5) \) on the graph.
2. **Use the Slope**:
- From the y-intercept \( (0, 5) \), use the slope \(-3\):
- Move down \( 3 \) units (because the slope is negative).
- Move right \( 1 \) unit.
- This brings you to the point \( (1, 2) \).
3. **Draw the Line**:
- After plotting the points \( (0, 5) \) and \( (1, 2) \), draw a straight line through these points.
### Additional Optional Points:
To further ensure accuracy and give more points to plot, you can continue applying the slope:
- From \( (1, 2) \):
- Move down \( 3 \) units and right \( 1 \) unit to arrive at \( (2, -1) \).
- Alternatively, moving in the opposite direction from \( (0, 5) \):
- Move up \( 3 \) units and left \( 1 \) unit to \( (-1, 8) \).
### Summary of Points to Plot:
- **Y-Intercept**: \( (0, 5) \)
- **Second Point using the slope**: \( (1, 2) \)
- **Additional Points**: \( (2, -1) \), \( (-1, 8) \)
### Summary of Identification:
- **Y-Intercept (b)**: \( 5 \)
- **Slope (m)**: \( -3 \)
### Example Graph Annotations:
- Label the axes:
- **X-Axis**: "x"
- **Y-Axis**: "y"
- Title the graph:
- **Title**: "Graph of the Linear Equation \( y - 5 = -3x \)"
- Label the points:
- "Y-Intercept (0, 5)"
- "Point (1, 2)"
- "Point (2, -1)"
Now you can graph the equation accurately with all the pertinent information labeled!
### Step 1: Identify the Slope and Y-Intercept
#### Rewrite in Slope-Intercept Form
The equation given is in a form that resembles point-slope form. To put it in slope-intercept form (\( y = mx + b \)), where \( m \) is the slope and \( b \) is the y-intercept, we can rearrange it:
\[
y - 5 = -3x
\]
Adding \( 5 \) to both sides:
\[
y = -3x + 5
\]
### Identification:
- **Y-Intercept (b)**: The y-intercept is \( 5 \). This means the line crosses the y-axis at the point \( (0, 5) \).
- **Slope (m)**: The slope is \( -3 \). This indicates that for every increase of \( 1 \) in \( x \), \( y \) decreases by \( 3 \) (a downward slope).
### Step 2: Graphing the Equation
1. **Plot the Y-Intercept**:
- Start at the point \( (0, 5) \) on the graph.
2. **Use the Slope**:
- From the y-intercept \( (0, 5) \), use the slope \(-3\):
- Move down \( 3 \) units (because the slope is negative).
- Move right \( 1 \) unit.
- This brings you to the point \( (1, 2) \).
3. **Draw the Line**:
- After plotting the points \( (0, 5) \) and \( (1, 2) \), draw a straight line through these points.
### Additional Optional Points:
To further ensure accuracy and give more points to plot, you can continue applying the slope:
- From \( (1, 2) \):
- Move down \( 3 \) units and right \( 1 \) unit to arrive at \( (2, -1) \).
- Alternatively, moving in the opposite direction from \( (0, 5) \):
- Move up \( 3 \) units and left \( 1 \) unit to \( (-1, 8) \).
### Summary of Points to Plot:
- **Y-Intercept**: \( (0, 5) \)
- **Second Point using the slope**: \( (1, 2) \)
- **Additional Points**: \( (2, -1) \), \( (-1, 8) \)
### Summary of Identification:
- **Y-Intercept (b)**: \( 5 \)
- **Slope (m)**: \( -3 \)
### Example Graph Annotations:
- Label the axes:
- **X-Axis**: "x"
- **Y-Axis**: "y"
- Title the graph:
- **Title**: "Graph of the Linear Equation \( y - 5 = -3x \)"
- Label the points:
- "Y-Intercept (0, 5)"
- "Point (1, 2)"
- "Point (2, -1)"
Now you can graph the equation accurately with all the pertinent information labeled!
Answered by
GPT-4o mini
Labeling a graph is important for clearly communicating the information it represents. Below are the steps you can follow to effectively label the graph of the equation \( y - 5 = -3x \), which has been transformed to slope-intercept form \( y = -3x + 5 \):
### Steps to Label the Graph:
1. **Title the Graph**:
- Write a descriptive title at the top of the graph. For example:
- **"Graph of the Linear Equation \( y - 5 = -3x \)"**
2. **Label the Axes**:
- On the horizontal axis (x-axis), write:
- **"x"**
- On the vertical axis (y-axis), write:
- **"y"**
3. **Identify and Label Key Points**:
- **Y-Intercept**: The point \( (0, 5) \) is where the line crosses the y-axis. You can label this point directly on the graph:
- **"Y-Intercept (0, 5)"** or simply put a small dot and write \( 5 \) next to it.
- **Other Points**: If you plotted additional points, such as \( (1, 2) \) and \( (2, -1) \), label them as well:
- **"Point (1, 2)"**
- **"Point (2, -1)"**
4. **Draw the Line**:
- Use a ruler to draw a straight line through the points you plotted. Make sure the line extends across the graph, indicating it continues indefinitely on both sides.
5. **Slope Indicator (Optional)**:
- You may want to indicate the slope on the graph. From the y-intercept \( (0, 5) \), you can use arrows to show the movement according to the slope \( -3 \) (down 3 units, right 1 unit):
- You can write:
- **"Slope = -3"**
- Draw an arrow from \( (0, 5) \) to \( (1, 2) \) and label it with this information.
6. **Grid (If applicable)**:
- If you are using graph paper, ensure that the grid is clear. Each intersection of the grid can correspond to unit increments for both axes.
7. **Ensure Clarity**:
- Use clear, legible text for all labels.
- Ensure that labels do not overlap the points or lines for better readability.
### Example Summary of Labels:
- **Title**: "Graph of the Linear Equation \( y - 5 = -3x \)"
- **X-Axis Label**: "x"
- **Y-Axis Label**: "y"
- **Point Labels**:
- "Y-Intercept (0, 5)"
- "Point (1, 2)"
- "Point (2, -1)"
- **Slope Indicator**: "Slope = -3"
By adhering to these guidelines, you can create a well-labeled and informative graph that effectively conveys the relationship represented by the linear equation.
### Steps to Label the Graph:
1. **Title the Graph**:
- Write a descriptive title at the top of the graph. For example:
- **"Graph of the Linear Equation \( y - 5 = -3x \)"**
2. **Label the Axes**:
- On the horizontal axis (x-axis), write:
- **"x"**
- On the vertical axis (y-axis), write:
- **"y"**
3. **Identify and Label Key Points**:
- **Y-Intercept**: The point \( (0, 5) \) is where the line crosses the y-axis. You can label this point directly on the graph:
- **"Y-Intercept (0, 5)"** or simply put a small dot and write \( 5 \) next to it.
- **Other Points**: If you plotted additional points, such as \( (1, 2) \) and \( (2, -1) \), label them as well:
- **"Point (1, 2)"**
- **"Point (2, -1)"**
4. **Draw the Line**:
- Use a ruler to draw a straight line through the points you plotted. Make sure the line extends across the graph, indicating it continues indefinitely on both sides.
5. **Slope Indicator (Optional)**:
- You may want to indicate the slope on the graph. From the y-intercept \( (0, 5) \), you can use arrows to show the movement according to the slope \( -3 \) (down 3 units, right 1 unit):
- You can write:
- **"Slope = -3"**
- Draw an arrow from \( (0, 5) \) to \( (1, 2) \) and label it with this information.
6. **Grid (If applicable)**:
- If you are using graph paper, ensure that the grid is clear. Each intersection of the grid can correspond to unit increments for both axes.
7. **Ensure Clarity**:
- Use clear, legible text for all labels.
- Ensure that labels do not overlap the points or lines for better readability.
### Example Summary of Labels:
- **Title**: "Graph of the Linear Equation \( y - 5 = -3x \)"
- **X-Axis Label**: "x"
- **Y-Axis Label**: "y"
- **Point Labels**:
- "Y-Intercept (0, 5)"
- "Point (1, 2)"
- "Point (2, -1)"
- **Slope Indicator**: "Slope = -3"
By adhering to these guidelines, you can create a well-labeled and informative graph that effectively conveys the relationship represented by the linear equation.
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