Question

Which systems of equations have no solutions? (Check all that apply.)
x – y = 1, –3x + 3y = 3
2y = 3 – 4x, x + y = 0
12x + 2 = 4y – 10, 2x + y = –11
–3x – y = 5, 15x = 10 – 5y
x + y = 1, –4x + 2y = 7

Answers

GPT-4o mini
To determine which systems of equations have no solutions, we look for systems that are inconsistent, meaning that the equations represent parallel lines. Parallel lines have the same slope but different y-intercepts, which means they will never intersect.

Let’s analyze each system of equations:

1. **System 1:**
- \( x - y = 1 \)
- \( -3x + 3y = 3 \)

Rearranging the second equation gives:
\[
3y = 3x + 3 \quad \Rightarrow \quad y = x + 1
\]

The first equation can be rearranged as:
\[
y = x - 1
\]

The slopes are both 1 (from \(y = x + 1\) and \(y = x - 1\)), therefore these lines are parallel. Since they have different y-intercepts (1 and -1), this system has **no solutions**.

2. **System 2:**
- \( 2y = 3 - 4x \)
- \( x + y = 0 \)

Rearranging the first equation:
\[
y = \frac{3}{2} - 2x
\]

Rearranging the second equation:
\[
y = -x
\]

The first equation has a slope of -2, and the second has a slope of -1. Thus, these lines are not parallel and will intersect at some point. Hence, this system has **solutions**.

3. **System 3:**
- \( 12x + 2 = 4y - 10 \)
- \( 2x + y = -11 \)

Rearranging the first equation gives:
\[
4y = 12x + 12 \quad \Rightarrow \quad y = 3x + 3
\]

Rearranging the second equation gives:
\[
y = -2x - 11
\]

The slopes are 3 and -2 respectively. These lines are not parallel and will intersect, meaning this system has **solutions**.

4. **System 4:**
- \( -3x - y = 5 \)
- \( 15x = 10 - 5y \)

Rearranging the second equation gives:
\[
5y = 10 - 15x \quad \Rightarrow \quad y = -3x + 2
\]

Rearranging the first equation gives:
\[
y = -3x - 5
\]

Both equations have the same slope of -3 but different intercepts (-5 and 2), making them parallel. Therefore, this system has **no solutions**.

5. **System 5:**
- \( x + y = 1 \)
- \( -4x + 2y = 7 \)

Rearranging the second equation gives:
\[
2y = 4x + 7 \quad \Rightarrow \quad y = 2x + \frac{7}{2}
\]

The first equation can be rearranged as:
\[
y = -x + 1
\]

The slopes are 2 and -1 respectively. These lines are not parallel and will intersect, indicating this system has **solutions**.

### Summary:
The systems of equations that have no solutions are:
- **System 1**: \( x - y = 1, -3x + 3y = 3 \)
- **System 4**: \( -3x - y = 5, 15x = 10 - 5y \)

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