Answer :
To determine how many solutions a given system of linear equations has, we follow these steps:
### System of Equations
Given the equations:
1. \( y = 4x + 4 \)
2. \( y = 4x - 1 \)
### Analyzing the Equations
1. Compare the slopes: Notice that both equations are written in the slope-intercept form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
- For \( y = 4x + 4 \), the slope (\( m \)) is 4.
- For \( y = 4x - 1 \), the slope (\( m \)) is also 4.
2. Compare the y-intercepts:
- In \( y = 4x + 4 \), the y-intercept is 4.
- In \( y = 4x - 1 \), the y-intercept is -1.
### Conclusion
- The slopes of the two lines are equal, which means the lines are parallel.
- Their y-intercepts are different, which means they do not coincide.
Since parallel lines that do not coincide will never intersect, this system of equations has:
No solution.
### System of Equations
Given the equations:
1. \( y = 4x + 4 \)
2. \( y = 4x - 1 \)
### Analyzing the Equations
1. Compare the slopes: Notice that both equations are written in the slope-intercept form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
- For \( y = 4x + 4 \), the slope (\( m \)) is 4.
- For \( y = 4x - 1 \), the slope (\( m \)) is also 4.
2. Compare the y-intercepts:
- In \( y = 4x + 4 \), the y-intercept is 4.
- In \( y = 4x - 1 \), the y-intercept is -1.
### Conclusion
- The slopes of the two lines are equal, which means the lines are parallel.
- Their y-intercepts are different, which means they do not coincide.
Since parallel lines that do not coincide will never intersect, this system of equations has:
No solution.
