To analyze the equations you've provided, let's first identify their characteristics.
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Equations:
- The first equation is \( y = \frac{2}{3}x + 1 \).
- The second equation is \( y = -\frac{2}{3}x + 1 \).
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Slope and Intercept:
- The first equation has a slope of \( \frac{2}{3} \) and a y-intercept of 1. This means the line rises as it goes from left to right.
- The second equation has a slope of \( -\frac{2}{3} \) and the same y-intercept of 1. This means the line falls as it goes from left to right.
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Graphing:
- Both lines intersect the y-axis at the point (0, 1).
- The first line, with a positive slope, will rise to the right, while the second line, with a negative slope, will fall to the right.
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Intersection Point:
- We can find the intersection point of these two lines by setting the equations equal to each other: \[ \frac{2}{3}x + 1 = -\frac{2}{3}x + 1 \]
- Simplifying this: \[ \frac{2}{3}x + \frac{2}{3}x = 1 - 1 \] \[ \frac{4}{3}x = 0 \quad \Rightarrow \quad x = 0 \]
- Plugging \( x = 0 \) back into either original equation gives: \[ y = \frac{2}{3}(0) + 1 = 1 \]
- Thus, the intersection point is (0, 1).
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Geometric Interpretation:
- The two lines are symmetrical around the y-axis and intersect at the point (0, 1). They diverge from this point and do not intersect again.
If you need further analysis, like finding points or graphing, please let me know!