Question
The given system of equations is graphed on the coordinate plane.
Which statements about the system are true? (Choose three.)
Responses
The solution to the system is the point of intersection between lines
.
The solution to the system is the point of intersection between lines e A n d f .
It is impossible for
to equal both
and
simultaneously; therefore, the system has no solution.
It is impossible for y to equal both 2 thirds x plus 1 and negative 3 halves x plus 4 simultaneously; therefore, the system has no solution.
The system is inconsistent because the lines are perpendicular.
The system is inconsistent because the lines are perpendicular.
The system is consistent because the lines are parallel.
The system is consistent because the lines are parallel.
Since the lines are perpendicular, the system has a solution.
Since the lines are perpendicular, the system has a solution.
The system is consistent because the lines
are perpendicular.
Which statements about the system are true? (Choose three.)
Responses
The solution to the system is the point of intersection between lines
.
The solution to the system is the point of intersection between lines e A n d f .
It is impossible for
to equal both
and
simultaneously; therefore, the system has no solution.
It is impossible for y to equal both 2 thirds x plus 1 and negative 3 halves x plus 4 simultaneously; therefore, the system has no solution.
The system is inconsistent because the lines are perpendicular.
The system is inconsistent because the lines are perpendicular.
The system is consistent because the lines are parallel.
The system is consistent because the lines are parallel.
Since the lines are perpendicular, the system has a solution.
Since the lines are perpendicular, the system has a solution.
The system is consistent because the lines
are perpendicular.
Answers
GPT-4o mini
To determine which statements are true about the system of equations and their relationships, let's analyze the implications of the different scenarios described:
1. **The solution to the system is the point of intersection between lines.**
- This statement is generally true for any system of linear equations; the solution is where their graphs intersect.
2. **The solution to the system is the point of intersection between lines e and f.**
- If lines e and f are the lines represented by the equations in the system, then this statement is also true and essentially reiterates the first point.
3. **It is impossible for (something) to equal both (something) and (something) simultaneously; therefore, the system has no solution.**
- This statement is true if the lines are parallel because parallel lines never intersect, resulting in no solution.
4. **It is impossible for y to equal both (2/3)x + 1 and (-3/2)x + 4 simultaneously; therefore, the system has no solution.**
- Again, this is true if the two equations describe parallel lines.
5. **The system is inconsistent because the lines are perpendicular.**
- This statement is false. If the lines are perpendicular, they will intersect at one point, meaning the system has exactly one solution, so it is consistent.
6. **The system is consistent because the lines are parallel.**
- This statement is false; parallel lines do not intersect, indicating that the system has no solution.
7. **Since the lines are perpendicular, the system has a solution.**
- This statement is true; perpendicular lines intersect, meaning the system has one solution.
8. **The system is consistent because the lines are perpendicular.**
- This statement is true for the same reason as above; if the lines are perpendicular, the system is consistent due to having one solution.
### True Statements
Based on the analysis, the true statements are:
1. The solution to the system is the point of intersection between lines.
2. The solution to the system is the point of intersection between lines e and f.
3. Since the lines are perpendicular, the system has a solution.
If you need further analysis or clarification, please provide the specific equations of the lines for more accurate conclusions.
1. **The solution to the system is the point of intersection between lines.**
- This statement is generally true for any system of linear equations; the solution is where their graphs intersect.
2. **The solution to the system is the point of intersection between lines e and f.**
- If lines e and f are the lines represented by the equations in the system, then this statement is also true and essentially reiterates the first point.
3. **It is impossible for (something) to equal both (something) and (something) simultaneously; therefore, the system has no solution.**
- This statement is true if the lines are parallel because parallel lines never intersect, resulting in no solution.
4. **It is impossible for y to equal both (2/3)x + 1 and (-3/2)x + 4 simultaneously; therefore, the system has no solution.**
- Again, this is true if the two equations describe parallel lines.
5. **The system is inconsistent because the lines are perpendicular.**
- This statement is false. If the lines are perpendicular, they will intersect at one point, meaning the system has exactly one solution, so it is consistent.
6. **The system is consistent because the lines are parallel.**
- This statement is false; parallel lines do not intersect, indicating that the system has no solution.
7. **Since the lines are perpendicular, the system has a solution.**
- This statement is true; perpendicular lines intersect, meaning the system has one solution.
8. **The system is consistent because the lines are perpendicular.**
- This statement is true for the same reason as above; if the lines are perpendicular, the system is consistent due to having one solution.
### True Statements
Based on the analysis, the true statements are:
1. The solution to the system is the point of intersection between lines.
2. The solution to the system is the point of intersection between lines e and f.
3. Since the lines are perpendicular, the system has a solution.
If you need further analysis or clarification, please provide the specific equations of the lines for more accurate conclusions.