Question

Which system of equations has exactly one solution?
2 x + y = 3. 5 x minus y = 11.
2 x + 2 y = 1. Negative 2 x minus 2 y = 1.
3 x + y = negative 1. 6 x + 2 y = negative 2.
3 x minus 2 y = 4. Negative 3 x + 2 y = 4.

Answer 1

To determine which system of equations has exactly one solution, we can analyze each of the given systems. A system of linear equations has exactly one solution if the lines represented by the equations intersect at exactly one point (which means the lines are not parallel).

### System 1:
1. \( 2x + y = 3 \)
2. \( 5x - y = 11 \)

To express both equations in slope-intercept form \( y = mx + b \):
1. \( y = -2x + 3 \) (Slope = -2)
2. \( -y = -5x + 11 \) ⇒ \( y = 5x - 11 \) (Slope = 5)

Since the slopes (-2 and 5) are different, the lines will intersect at exactly one point. **This system has exactly one solution.**

### System 2:
1. \( 2x + 2y = 1 \)
2. \( -2x - 2y = 1 \)

Rearranging the second equation gives us:
\( 2x + 2y = -1 \)

Now both equations can be written as:
1. \( 2x + 2y = 1 \)
2. \( 2x + 2y = -1 \)

These lines are parallel because they have the same coefficients for \( x \) and \( y \) and different constants. **This system has no solutions.**

### System 3:
1. \( 3x + y = -1 \)
2. \( 6x + 2y = -2 \)

The second equation can be simplified by dividing everything by 2:
\( 3x + y = -1 \)

Both equations are identical; thus, they represent the same line. **This system has infinitely many solutions.**

### System 4:
1. \( 3x - 2y = 4 \)
2. \( -3x + 2y = 4 \)

Rearranging the second equation gives:
\( 3x - 2y = -4 \)

Now we have:
1. \( 3x - 2y = 4 \)
2. \( 3x - 2y = -4 \)

These are parallel lines (same slope, different y-intercepts). **This system has no solutions.**

### Conclusion:
The only system that has exactly one solution is **System 1**:
\[ 2x + y = 3 \]
\[ 5x - y = 11 \]