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:
A. consistent independent
B. consistent dependent
C. inconsistent

This means the system has:
A. a unique solution
B. infinitely many solutions
C. no solution



Answer :

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