To determine the nature of the system of linear equations represented by the given lines, we analyze the equations step-by-step:
1. The given lines are:
- Line 1: [tex]\( y = \frac{1}{2} x - 1 \)[/tex]
- Line 2: [tex]\( y = \frac{1}{2} x + 4 \)[/tex]
2. The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Compare the slopes ([tex]\( m \)[/tex]) of both lines:
- The slope of Line 1 is [tex]\( \frac{1}{2} \)[/tex].
- The slope of Line 2 is [tex]\( \frac{1}{2} \)[/tex].
4. Observe that both lines have the same slope, [tex]\( \frac{1}{2} \)[/tex], meaning the lines are parallel.
5. Parallel lines do not intersect unless they are identical. Since the y-intercepts are different ([tex]\(-1\)[/tex] for Line 1 and [tex]\(4\)[/tex] for Line 2), the lines are not the same.
6. Because parallel lines with different y-intercepts do not intersect, there are no common points between the lines.
Therefore, the system of equations is identified as:
- Inconsistent: This means the lines do not intersect, indicating the system does not have any solutions.
Consequently, this means the system has:
- No solution: There are no points that satisfy both equations simultaneously.
The answers to fill in the given solution are:
System A
Line 1: [tex]\( y = \frac{1}{2} x - 1 \)[/tex]
Line 2: [tex]\( y = \frac{1}{2} x + 4 \)[/tex]
This system of equations is:
- inconsistent
This means the system has:
- no solution
