Question
Use the table to answer the question.
Table of Values
x ya
yb
0 6 −3
1 7 −1
2 8 1
Which of the following systems of linear equations is represented by the table of values?
(1 point)
Responses
{y=x−3y=2x+6
{y=x+6y=2x−3
{y=−x+6y=−2x−3
{y=x−6y=2x+3
Table of Values
x ya
yb
0 6 −3
1 7 −1
2 8 1
Which of the following systems of linear equations is represented by the table of values?
(1 point)
Responses
{y=x−3y=2x+6
{y=x+6y=2x−3
{y=−x+6y=−2x−3
{y=x−6y=2x+3
Answers
Answer
Use the image to answer the question.
Two lines intersect on the coordinate plane. The x axis ranges from negative 10 to 5 in increments of 5. The y axis ranges from negative 5 to 10 in increments of 5.
What are the equations for the system of equations graphed?
(1 point)
Responses
{y=12x+7y=−52x−5
enlarged left brace start layout first row y equals one half x plus 7 second row y equals negative five halves x minus 5 end layout
{y=−12x+7y=52x−5
enlarged left brace start layout first row y equalsnegative one half x plus 7 second row y equals five halves x minus 5 end layout
{y=2x+7y=−52x−5
enlarged left brace start layout first row y equals 2 x plus 7 second row y equals negative five halves x minus 5 end layout
{y=12x+7y=−5x−5
Two lines intersect on the coordinate plane. The x axis ranges from negative 10 to 5 in increments of 5. The y axis ranges from negative 5 to 10 in increments of 5.
What are the equations for the system of equations graphed?
(1 point)
Responses
{y=12x+7y=−52x−5
enlarged left brace start layout first row y equals one half x plus 7 second row y equals negative five halves x minus 5 end layout
{y=−12x+7y=52x−5
enlarged left brace start layout first row y equalsnegative one half x plus 7 second row y equals five halves x minus 5 end layout
{y=2x+7y=−52x−5
enlarged left brace start layout first row y equals 2 x plus 7 second row y equals negative five halves x minus 5 end layout
{y=12x+7y=−5x−5
Answered by
GPT-4o mini
To determine which system of linear equations is represented by the given table of values, we can analyze the points provided.
The table has the following data points:
- (0, 6)
- (1, 7)
- (2, 8)
These points suggest a linear relationship, which we can investigate by finding the equation of the line that passes through them.
1. **Finding the slope (m):**
The slope (m) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using points (0, 6) and (1, 7):
\[
m = \frac{7 - 6}{1 - 0} = \frac{1}{1} = 1
\]
2. **Finding the y-intercept (b):**
We can use the slope-intercept form of the equation y = mx + b. We know m = 1, and we can use the point (0, 6) to find b:
\[
6 = 1(0) + b \implies b = 6
\]
Thus, the first equation is:
\[
y = x + 6
\]
3. **Finding another equation:**
Let's use points (1, 7) and (2, 8) to find the equation of another line.
- The slope is:
\[
m = \frac{8 - 7}{2 - 1} = \frac{1}{1} = 1
\]
- Using point (1, 7):
\[
7 = 1(1) + b \implies b = 6
\]
The second equation remains consistent:
\[
y = x + 6
\]
Now, we need to derive another equation that could possibly represent the table of values. The next two equations can either be contradictory or dependent. Each choice states a unique second equation:
1. **Choice \( \{ y = x - 3; y = 2x + 6 \} \)**
2. **Choice \( \{ y = x + 6; y = 2x - 3 \} \)**
3. **Choice \( \{ y = -x + 6; y = -2x - 3 \} \)**
4. **Choice \( \{ y = x - 6; y = 2x + 3 \} \)**
Since one confirmed equation is \( y = x + 6 \), let's check the second equation from each choice:
- **Choice 2, \( y = 2x - 3 \)**:
If we use this equation with \( x = 0 \):
\[
y = 2(0) - 3 = -3 \quad \text{(not in the table)}
\]
Check \( x = 1 \):
\[
y = 2(1) - 3 = -1 \quad \text{(not in the table)}
\]
Check \( x = 2 \):
\[
y = 2(2) - 3 = 1 \quad \text{(not in the table)}
\]
After careful analysis, since \( y = x - 3 \) from choice 1 contradicts our earlier derived equations, and the remaining choices don't match up consistently, we can conclude:
Given the table of values, the system that includes \( y = x + 6 \) must match the second system set to represent consistent mappings.
Thus the answer is:
**{y = x + 6, y = 2x - 3}**.
The table has the following data points:
- (0, 6)
- (1, 7)
- (2, 8)
These points suggest a linear relationship, which we can investigate by finding the equation of the line that passes through them.
1. **Finding the slope (m):**
The slope (m) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using points (0, 6) and (1, 7):
\[
m = \frac{7 - 6}{1 - 0} = \frac{1}{1} = 1
\]
2. **Finding the y-intercept (b):**
We can use the slope-intercept form of the equation y = mx + b. We know m = 1, and we can use the point (0, 6) to find b:
\[
6 = 1(0) + b \implies b = 6
\]
Thus, the first equation is:
\[
y = x + 6
\]
3. **Finding another equation:**
Let's use points (1, 7) and (2, 8) to find the equation of another line.
- The slope is:
\[
m = \frac{8 - 7}{2 - 1} = \frac{1}{1} = 1
\]
- Using point (1, 7):
\[
7 = 1(1) + b \implies b = 6
\]
The second equation remains consistent:
\[
y = x + 6
\]
Now, we need to derive another equation that could possibly represent the table of values. The next two equations can either be contradictory or dependent. Each choice states a unique second equation:
1. **Choice \( \{ y = x - 3; y = 2x + 6 \} \)**
2. **Choice \( \{ y = x + 6; y = 2x - 3 \} \)**
3. **Choice \( \{ y = -x + 6; y = -2x - 3 \} \)**
4. **Choice \( \{ y = x - 6; y = 2x + 3 \} \)**
Since one confirmed equation is \( y = x + 6 \), let's check the second equation from each choice:
- **Choice 2, \( y = 2x - 3 \)**:
If we use this equation with \( x = 0 \):
\[
y = 2(0) - 3 = -3 \quad \text{(not in the table)}
\]
Check \( x = 1 \):
\[
y = 2(1) - 3 = -1 \quad \text{(not in the table)}
\]
Check \( x = 2 \):
\[
y = 2(2) - 3 = 1 \quad \text{(not in the table)}
\]
After careful analysis, since \( y = x - 3 \) from choice 1 contradicts our earlier derived equations, and the remaining choices don't match up consistently, we can conclude:
Given the table of values, the system that includes \( y = x + 6 \) must match the second system set to represent consistent mappings.
Thus the answer is:
**{y = x + 6, y = 2x - 3}**.
Answered by
GPT-4o mini
To determine the equations for the lines graphed in the coordinate plane, we need to identify the slopes and y-intercepts of each line from the graph. However, without the visual image to refer to, I can guide you through how you would solve this problem based on the information typically required:
1. **Identify the y-intercept (b)** of each line (the point where the line crosses the y-axis).
2. **Determine the slope (m)**, which is calculated as the change in y over the change in x (rise/run).
After having these two pieces of information, you can construct the equation for each line in the format of:
\[
y = mx + b
\]
Once you have the equations for both lines, you would then match those equations with the options given:
1. **{y = 12x + 7, y = -52x - 5}**
2. **{y = \frac{1}{2}x + 7, y = -\frac{5}{2}x - 5}**
3. **{y = 2x + 7, y = -\frac{5}{2}x - 5}**
4. **{y = 12x + 7, y = -5x - 5}**
**Steps to Analyze:**
- Check if you can see where the lines intersects the y-axis (this will give you the y-intercept).
- Look closely at the slope by selecting two points on each line from the graph. Calculate the slope for both lines.
- Substitute these values into the line equation format and write down both equations.
Once you derive the equations, compare them with those in the response options to find the correct matching system of equations.
If you can provide specific points or descriptions from the image, I'd be able to help calculate the exact equations for each line.
1. **Identify the y-intercept (b)** of each line (the point where the line crosses the y-axis).
2. **Determine the slope (m)**, which is calculated as the change in y over the change in x (rise/run).
After having these two pieces of information, you can construct the equation for each line in the format of:
\[
y = mx + b
\]
Once you have the equations for both lines, you would then match those equations with the options given:
1. **{y = 12x + 7, y = -52x - 5}**
2. **{y = \frac{1}{2}x + 7, y = -\frac{5}{2}x - 5}**
3. **{y = 2x + 7, y = -\frac{5}{2}x - 5}**
4. **{y = 12x + 7, y = -5x - 5}**
**Steps to Analyze:**
- Check if you can see where the lines intersects the y-axis (this will give you the y-intercept).
- Look closely at the slope by selecting two points on each line from the graph. Calculate the slope for both lines.
- Substitute these values into the line equation format and write down both equations.
Once you derive the equations, compare them with those in the response options to find the correct matching system of equations.
If you can provide specific points or descriptions from the image, I'd be able to help calculate the exact equations for each line.
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