To analyze the system of equations given by the lines \( e: y = 3x + 2 \) and \( f: y = -\frac{1}{3}x + 2 \), we can evaluate the characteristics of the lines and their intersection.
- Finding the slopes:
- Line \( e \) has a slope of \( 3 \).
- Line \( f \) has a slope of \( -\frac{1}{3} \).
Since the slopes are different, the lines are neither parallel nor identical; they will intersect at some point.
- Finding the intersection point: To find the point of intersection, we can set the two equations equal to each other:
\[ 3x + 2 = -\frac{1}{3}x + 2 \]
Subtract \( 2 \) from both sides:
\[ 3x = -\frac{1}{3}x \]
Combine like terms:
\[ 3x + \frac{1}{3}x = 0 \]
Multiply every term by \( 3 \) to eliminate the fraction:
\[ 9x + x = 0 \]
Combine:
\[ 10x = 0 \]
So \( x = 0 \). Now, substituting \( x = 0 \) back into either equation to find \( y \):
Using line \( e \):
\[ y = 3(0) + 2 = 2 \]
Thus, the solution to the system is \( (0, 2) \).
- Conclusion about the statements:
- The solution to the system is (0, 2): This statement is true.
- It is impossible for y to equal both 3x + 2 and -1/3x + 2 simultaneously; therefore, the system has no solution: This statement is false because the system does have a solution.
- The system is inconsistent because the lines are perpendicular: This statement is false; the system is consistent since the lines intersect.
- The system is consistent because the lines are parallel: This statement is false; the lines are not parallel.
- Since the lines are perpendicular, the system has a solution: This statement is true; perpendicular lines always intersect, meaning there is a solution.
- The system is consistent because the lines e and f are perpendicular: This statement is also true.
Based on this analysis, the three correct statements would be:
- The solution to the system is (0, 2).
- Since the lines are perpendicular, the system has a solution.
- The system is consistent because the lines e and f are perpendicular.