Let's analyze the given lines \( e \) and \( f \).
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Lines:
- Line \( e: y = 3x + 2 \) has a slope of 3.
- Line \( f: y = -\frac{1}{3}x + 2 \) has a slope of \(-\frac{1}{3}\).
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Identifying Relationships:
- The slopes of the two lines are different, which means they are not parallel.
- The product of the slopes \((3) \cdot (-\frac{1}{3}) = -1\), indicating that the lines are perpendicular.
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Intersection Point: To find the solution (intersection point), we can set the equations equal to each other: \[ 3x + 2 = -\frac{1}{3}x + 2 \] Subtract 2 from both sides: \[ 3x = -\frac{1}{3}x \] Multiply both sides by 3 to eliminate the fraction: \[ 9x = -x \] Add \( x \) to both sides: \[ 10x = 0 \Rightarrow x = 0 \]
Substitute \( x = 0 \) back into either equation to find \( y \): \[ y = 3(0) + 2 = 2 \] So the solution to the system is \( (0, 2) \).
Based on this analysis, here are the true statements regarding the system:
- The solution to the system is (0, 2).
- Since the lines are perpendicular, the system has a solution.
- *It is impossible for \( y \) to equal both \( 3x + 2 \) and \(-\frac{1}{3}x + 2\) simultaneously; therefore, the system has no solution. (This statement is not true. The system has a solution.)
From the options you provided, the following three true statements should be chosen:
- The solution to the system is (0, 2).
- Since the lines are perpendicular, the system has a solution.
- (A duplicated option, but to note) - The correct interpretation regarding lines being perpendicular leading to a consistent system with a solution.
(Note: You should confirm the wording of the responses as there's repeated wording around inconsistency which may indicate errors in the provided options.)
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