To graph the system of equations \(y = 3x + 5\) and \(y = 3x + 2\), we will first determine the characteristics of each line:
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Slope-Intercept Form: Both equations are in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- For \(y = 3x + 5\): The slope is 3, and the y-intercept is 5, meaning it crosses the y-axis at (0, 5).
- For \(y = 3x + 2\): The slope is also 3, and the y-intercept is 2, meaning it crosses the y-axis at (0, 2).
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Graphing the Lines:
- For the equation \(y = 3x + 5\): Start at (0, 5) and use the slope (rise/run = 3/1) to find another point. From (0, 5), move up 3 units and right 1 unit to reach (1, 8).
- For the equation \(y = 3x + 2\): Start at (0, 2) and again use the slope to find another point. From (0, 2), move up 3 units and right 1 unit to reach (1, 5).
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Characteristics of the Lines:
- Both lines have the same slope (3) but different y-intercepts (5 and 2).
Conclusion
Since the lines are parallel (they never intersect), there are no points that satisfy both equations simultaneously.
Number of Solutions: The system has no solutions.
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