Question

Which of the following systems of linear equations has no solution?

A: | 4x - y = 17
| 4x - y = 3

B: |4x - y = -17
| 4x + y = 3

C: | 4x - y = 17
| 4x + 7 = 3

D: | 4x + y = 17
| 4x - y = 3

Answer 1

To determine which system of linear equations has no solution, we must examine each pair of equations to see if they are parallel (having the same slope but different y-intercepts).

### A:
Equations:
1. \( 4x - y = 17 \)
2. \( 4x - y = 3 \)

Both equations have the same slope (4) but different y-intercepts, which means they are parallel. Therefore, **A has no solution**.

### B:
Equations:
1. \( 4x - y = -17 \)
2. \( 4x + y = 3 \)

Rearranging the second equation gives \( y = -4x + 3 \). The slope of the first equation is 4, and the second equation’s slope is -4; since they have different slopes, they will intersect at one point. Thus, **B has a solution**.

### C:
Equations:
1. \( 4x - y = 17 \)
2. \( 4x + 7 = 3 \)

First equation is unchanged. Rearranging the second gives:

\( 4x = 3 - 7 \)
\( 4x = -4 \)
\( x = -1 \)

We can find \( y \) from the first equation:

Plugging \( x = -1 \) into \( 4x - y = 17 \):

\( 4(-1) - y = 17 \)
\( -4 - y = 17 \)
\( -y = 21 \)
\( y = -21 \)

Thus, **C has a solution**.

### D:
Equations:
1. \( 4x + y = 17 \)
2. \( 4x - y = 3 \)

Rearranging both equations:
From the first equation, we can express y:
\( y = 17 - 4x \)
From the second equation:
\( -y = 3 - 4x \) or \( y = 4x - 3 \)

These two equations have different slopes, and thus they will intersect at one point. Therefore, **D has a solution**.

### Conclusion:
The system of linear equations that has no solution is **A**.