Question
Which of the graphs with no solution correctly shows the system of equations y=2x+4 and −2x+y=8 ?(1 point)
Responses
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 0 comma 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 2 comma 0 right parenthesis and left parenthesis 0 comma 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Responses
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 0 comma 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 2 comma 0 right parenthesis and left parenthesis 0 comma 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Answers
Answer
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Answered by
GPT-4o mini
To determine which of the three graphs illustrates a system of equations with no solution for \(y = 2x + 4\) and \(−2x + y = 8\), we first need to explore the relationships represented by the equations.
1. **Equation Analysis**:
- The first equation, \(y = 2x + 4\), is in slope-intercept form \(y = mx + b\), where \(m = 2\) and \(b = 4\). This means the line has a slope of 2 and a y-intercept at (0, 4).
- The second equation, \(−2x + y = 8\), can be rewritten in slope-intercept form as \(y = 2x + 8\), where \(m = 2\) and \(b = 8\). This line also has a slope of 2 but a y-intercept at (0, 8).
2. **Interpretation of the Equations**:
- Both equations have the same slope (2), which means they are parallel. Since they have different y-intercepts (4 and 8), they will never intersect. Therefore, this system of equations has no solution.
3. **Graphic Representation**:
- From the description of the graphs provided:
- The correct graph must show two parallel lines. Since both equations have a slope of 2, the lines that represent these equations must be parallel.
- Additionally, the lines must be consistently offset from one another to reflect their different y-intercepts (which means one line is consistently above the other).
4. **Evaluating the Graphs**:
- The first image indicates a line passing through (0, 4) and (−2, 0) (the solid line) and another line passing through (−4, 0) and (0, 8) (the dashed line).
- The second image indicates a solid line through (−2, 0) and (0, −4) with a dashed line through (−4, 0) and (0, −8) (these do not both have the correct y-intercepts).
- The third image indicates a solid line through (2, 0) and (0, 4) and a dashed line through (−4, 0) and (0, −8) (again, this does not maintain the right intercept structure).
Given this analysis, the first image appears to best depict the scenario described, showing parallel lines with different y-intercepts. Thus, the answer is likely the first option.
1. **Equation Analysis**:
- The first equation, \(y = 2x + 4\), is in slope-intercept form \(y = mx + b\), where \(m = 2\) and \(b = 4\). This means the line has a slope of 2 and a y-intercept at (0, 4).
- The second equation, \(−2x + y = 8\), can be rewritten in slope-intercept form as \(y = 2x + 8\), where \(m = 2\) and \(b = 8\). This line also has a slope of 2 but a y-intercept at (0, 8).
2. **Interpretation of the Equations**:
- Both equations have the same slope (2), which means they are parallel. Since they have different y-intercepts (4 and 8), they will never intersect. Therefore, this system of equations has no solution.
3. **Graphic Representation**:
- From the description of the graphs provided:
- The correct graph must show two parallel lines. Since both equations have a slope of 2, the lines that represent these equations must be parallel.
- Additionally, the lines must be consistently offset from one another to reflect their different y-intercepts (which means one line is consistently above the other).
4. **Evaluating the Graphs**:
- The first image indicates a line passing through (0, 4) and (−2, 0) (the solid line) and another line passing through (−4, 0) and (0, 8) (the dashed line).
- The second image indicates a solid line through (−2, 0) and (0, −4) with a dashed line through (−4, 0) and (0, −8) (these do not both have the correct y-intercepts).
- The third image indicates a solid line through (2, 0) and (0, 4) and a dashed line through (−4, 0) and (0, −8) (again, this does not maintain the right intercept structure).
Given this analysis, the first image appears to best depict the scenario described, showing parallel lines with different y-intercepts. Thus, the answer is likely the first option.
Answered by
GPT-4o mini
Let’s double-check the equations and analyze the scenario one more time for clarity.
1. **Equations**:
- The first equation is \(y = 2x + 4\).
- The second equation can be rewritten as follows:
\[
-2x + y = 8 \implies y = 2x + 8
\]
2. **Slope and Intercepts**:
- Both equations have the same slope of **2**.
- The first equation has a y-intercept of **4** (the line crosses the y-axis at \( (0, 4) \)).
- The second equation has a y-intercept of **8** (the line crosses the y-axis at \( (0, 8) \)).
3. **Relationship**:
- Since both lines have the same slope but different intercepts, they are parallel and will never intersect. This means there is no solution to the system.
4. **Graph Analysis Based on Your Descriptions**:
- **First Image**: Solid line through (0, 4) and (−2, 0) with a dashed line through (−4, 0) and (0, 8)—correctly matches y-intercepts (4 and 8), so these are two parallel lines.
- **Second Image**: Solid line through (−2, 0) and (0, −4) does not correspond to either equation; therefore, incorrect.
- **Third Image**: Similar situation; the lines represented do not have the correct slopes or intercepts.
### Conclusion
The first image accurately represents the system of equations with no solution, as it properly displays parallel lines with the correct relationships relative to their y-intercepts. If you're asking whether I was correct in my initial conclusion—the first choice is indeed the correct graph illustrating the described system of equations having no solution.
1. **Equations**:
- The first equation is \(y = 2x + 4\).
- The second equation can be rewritten as follows:
\[
-2x + y = 8 \implies y = 2x + 8
\]
2. **Slope and Intercepts**:
- Both equations have the same slope of **2**.
- The first equation has a y-intercept of **4** (the line crosses the y-axis at \( (0, 4) \)).
- The second equation has a y-intercept of **8** (the line crosses the y-axis at \( (0, 8) \)).
3. **Relationship**:
- Since both lines have the same slope but different intercepts, they are parallel and will never intersect. This means there is no solution to the system.
4. **Graph Analysis Based on Your Descriptions**:
- **First Image**: Solid line through (0, 4) and (−2, 0) with a dashed line through (−4, 0) and (0, 8)—correctly matches y-intercepts (4 and 8), so these are two parallel lines.
- **Second Image**: Solid line through (−2, 0) and (0, −4) does not correspond to either equation; therefore, incorrect.
- **Third Image**: Similar situation; the lines represented do not have the correct slopes or intercepts.
### Conclusion
The first image accurately represents the system of equations with no solution, as it properly displays parallel lines with the correct relationships relative to their y-intercepts. If you're asking whether I was correct in my initial conclusion—the first choice is indeed the correct graph illustrating the described system of equations having no solution.
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