Question

Are these lines parallel?
2x+2y=2 and 2y=-2x+3
Y=3x+2 and y+3x=-5
Y=2x-3 and 6x+3y-9=0
Y-2x+6=0 and 4y=8x+1

Answer 1

To determine if the given lines are parallel, we need to rewrite each equation in the slope-intercept form (y = mx + b), where m represents the slope. If two lines have the same slope, they are parallel.

Let's analyze each pair of lines:

1. **Lines: \(2x + 2y = 2\) and \(2y = -2x + 3\)**

First equation:
\[
2y = -2x + 2 \implies y = -x + 1 \quad \text{(slope = -1)}
\]

Second equation:
\[
2y = -2x + 3 \implies y = -x + \frac{3}{2} \quad \text{(slope = -1)}
\]

Both lines have the same slope \(-1\), so they are parallel.

2. **Lines: \(y = 3x + 2\) and \(y + 3x = -5\)**

First equation:
\[
y = 3x + 2 \quad \text{(slope = 3)}
\]

Second equation:
\[
y = -3x - 5 \quad \text{(slope = -3)}
\]

The slopes are different (3 and -3), so they are not parallel.

3. **Lines: \(y = 2x - 3\) and \(6x + 3y - 9 = 0\)**

First equation:
\[
y = 2x - 3 \quad \text{(slope = 2)}
\]

Second equation:
\[
3y = -6x + 9 \implies y = -2x + 3 \quad \text{(slope = -2)}
\]

The slopes are different (2 and -2), so they are not parallel.

4. **Lines: \(y - 2x + 6 = 0\) and \(4y = 8x + 1\)**

First equation:
\[
y = 2x - 6 \quad \text{(slope = 2)}
\]

Second equation:
\[
y = 2x + \frac{1}{4} \quad \text{(slope = 2)}
\]

Both lines have the same slope \(2\), so they are parallel.

### Summary:
- The lines \(2x + 2y = 2\) and \(2y = -2x + 3\) are parallel.
- The lines \(y = 3x + 2\) and \(y + 3x = -5\) are not parallel.
- The lines \(y = 2x - 3\) and \(6x + 3y - 9 = 0\) are not parallel.
- The lines \(y - 2x + 6 = 0\) and \(4y = 8x + 1\) are parallel.