Question
(a)
Graph the system of linear equations and then state how many solutions
y=3x+5
y=3x+2
Answers
GPT-4o mini
To graph the system of equations \(y = 3x + 5\) and \(y = 3x + 2\), we will first determine the characteristics of each line:
1. **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).
2. **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).
3. **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**.
1. **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).
2. **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).
3. **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**.