To determine how many solutions the system of linear equations has, we start by analyzing the two equations provided:
- \( y = 3x + 5 \)
- \( y = 3x + 2 \)
Both equations are in slope-intercept form \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept.
Step 1: Identify the slopes and y-intercepts
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For the first equation \( y = 3x + 5 \):
- Slope (m) = 3
- y-intercept (b) = 5
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For the second equation \( y = 3x + 2 \):
- Slope (m) = 3
- y-intercept (b) = 2
Step 2: Analyze the lines
- Both lines have the same slope (3), which means they are parallel.
- Since the y-intercepts are different (5 for the first line and 2 for the second line), the lines will never intersect.
Conclusion
Since the lines are parallel, there is no point of intersection between them, and therefore:
- The system of linear equations has no solutions.
Graphing
If you were to graph these two lines:
- The line \( y = 3x + 5 \) would cross the y-axis at the point (0, 5).
- The line \( y = 3x + 2 \) would cross the y-axis at the point (0, 2).
Both lines would increase at the same rate due to the slope of 3 but would never meet, confirming that there are no solutions.