To analyze the two lines given by the equations:
- Line e: \( y = -\frac{1}{3}x + 6 \)
- Line f: \( y = 4x - 7 \)
Step 1: Determine the slopes
- For line e, the slope is \( -\frac{1}{3} \).
- For line f, the slope is \( 4 \).
Step 2: Identify if the lines are parallel or perpendicular
- The lines are not parallel since they have different slopes.
- The lines are not perpendicular either, as the product of their slopes \( (-\frac{1}{3}) \times 4 = -\frac{4}{3} \) is not equal to -1.
Step 3: Check for an intersection
- Since the lines are not parallel, they will intersect at one point.
Step 4: Find the intersection point
To find the intersection point, set the equations equal: \[ -\frac{1}{3}x + 6 = 4x - 7 \]
Multiply through by 3 to eliminate the fraction: \[ -x + 18 = 12x - 21 \]
Combine like terms: \[ 18 + 21 = 12x + x \] \[ 39 = 13x \] \[ x = 3 \]
Now substitute \( x = 3 \) back into either original equation to find \( y \): Using line f: \[ y = 4(3) - 7 = 12 - 7 = 5 \]
Thus, the intersection point (solution to the system) is \( (3, 5) \).
Conclusions
Now we can evaluate the statements:
- The solution to the system is (3,5) - True
- It is impossible for \( y \) to equal both \( 4x - 7 \) and \( -\frac{1}{3}x + 6 \) simultaneously; therefore, the system has no solution. - False
- The system is consistent because the lines are perpendicular. - False
- The system is consistent because the lines are NOT parallel. - True
- The system is inconsistent because the lines e and f intersect. - False
Final Answers
The two true statements are:
- The solution to the system is (3,5).
- The system is consistent because the lines are NOT parallel.
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