Match parallel lines from equations in four grids—each grid has one unmatched equation.

To solve the problem of matching the lines that are parallel in each grid, we need to identify equations with the same slope. Recall that two lines are parallel if and only if their slopes are equal. Let's go through each grid step by step.

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Grid 1

The equations are:
1. \( y = 2x + 3 \)
2. \( y = 2 - 3x \)
3. \( 2y = 6x + 6 \)
4. \( y + 3x = 2 \)
5. \( y - 2x = 3 \)
6. \( 2y = x + 3 \)
7. \( y = 3x + 3 \)
8. \( y = 2x + 2 \)
9. \( y = \frac{x}{2} + \frac{3}{2} \)

#### Step 1: Rewrite each equation in slope-intercept form \( y = mx + b \)
1. \( y = 2x + 3 \) → slope \( m = 2 \)
2. \( y = 2 - 3x \) → rewrite as \( y = -3x + 2 \) → slope \( m = -3 \)
3. \( 2y = 6x + 6 \) → divide by 2: \( y = 3x + 3 \) → slope \( m = 3 \)
4. \( y + 3x = 2 \) → rewrite as \( y = -3x + 2 \) → slope \( m = -3 \)
5. \( y - 2x = 3 \) → rewrite as \( y = 2x + 3 \) → slope \( m = 2 \)
6. \( 2y = x + 3 \) → divide by 2: \( y = \frac{x}{2} + \frac{3}{2} \) → slope \( m = \frac{1}{2} \)
7. \( y = 3x + 3 \) → slope \( m = 3 \)
8. \( y = 2x + 2 \) → slope \( m = 2 \)
9. \( y = \frac{x}{2} + \frac{3}{2} \) → slope \( m = \frac{1}{2} \)

#### Step 2: Group equations by slope
- Slope \( m = 2 \): \( y = 2x + 3 \), \( y = 2x + 2 \), \( y - 2x = 3 \)
- Slope \( m = -3 \): \( y = 2 - 3x \), \( y + 3x = 2 \)
- Slope \( m = 3 \): \( 2y = 6x + 6 \), \( y = 3x + 3 \)
- Slope \( m = \frac{1}{2} \): \( 2y = x + 3 \), \( y = \frac{x}{2} + \frac{3}{2} \)

#### Step 3: Identify the leftover equation
The equation \( y = \frac{x}{2} + \frac{3}{2} \) is already matched with \( 2y = x + 3 \). The leftover equation is:
\[ \boxed{y = 3x + 3} \]

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Grid 2

The equations are:
1. \( y = \frac{1}{2}x + 1 \)
2. \( y + x - 5 = 0 \)
3. \( y = 5 - x \)
4. \( y + x = 5 \)
5. \( y = 5 \)
6. \( 5 - y = 0 \)
7. \( 2y = 5x + 3 \)
8. \( 2y = x + 2 \)
9. \( 2y - 5x = 3 \)

#### Step 1: Rewrite each equation in slope-intercept form \( y = mx + b \)
1. \( y = \frac{1}{2}x + 1 \) → slope \( m = \frac{1}{2} \)
2. \( y + x - 5 = 0 \) → rewrite as \( y = -x + 5 \) → slope \( m = -1 \)
3. \( y = 5 - x \) → rewrite as \( y = -x + 5 \) → slope \( m = -1 \)
4. \( y + x = 5 \) → rewrite as \( y = -x + 5 \) → slope \( m = -1 \)
5. \( y = 5 \) → horizontal line, slope \( m = 0 \)
6. \( 5 - y = 0 \) → rewrite as \( y = 5 \) → slope \( m = 0 \)
7. \( 2y = 5x + 3 \) → divide by 2: \( y = \frac{5}{2}x + \frac{3}{2} \) → slope \( m = \frac{5}{2} \)
8. \( 2y = x + 2 \) → divide by 2: \( y = \frac{1}{2}x + 1 \) → slope \( m = \frac{1}{2} \)
9. \( 2y - 5x = 3 \) → rewrite as \( 2y = 5x + 3 \) → divide by 2: \( y = \frac{5}{2}x + \frac{3}{2} \) → slope \( m = \frac{5}{2} \)

#### Step 2: Group equations by slope
- Slope \( m = \frac{1}{2} \): \( y = \frac{1}{2}x + 1 \), \( 2y = x + 2 \)
- Slope \( m = -1 \): \( y + x - 5 = 0 \), \( y = 5 - x \), \( y + x = 5 \)
- Slope \( m = 0 \): \( y = 5 \), \( 5 - y = 0 \)
- Slope \( m = \frac{5}{2} \): \( 2y = 5x + 3 \), \( 2y - 5x = 3 \)

#### Step 3: Identify the leftover equation
The equation \( y = 5 \) is already matched with \( 5 - y = 0 \). The leftover equation is:
\[ \boxed{y = 5} \]

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Grid 3

The equations are:
1. \( y = 1.5x + 3 \)
2. \( y = 0.25x \)
3. \( y = \frac{1}{4}x - \frac{3}{10} \)
4. \( y = 1.3x - 1 \)
5. \( y = 0.3x + 1 \)
6. \( y = \frac{8}{5}x - 1 \)
7. \( y = \frac{3}{10}x \)
8. \( y = 1.6x + 7 \)
9. \( y = \frac{3}{2}x - 6 \)

#### Step 1: Identify the slopes
1. \( y = 1.5x + 3 \) → slope \( m = 1.5 \)
2. \( y = 0.25x \) → slope \( m = 0.25 \)
3. \( y = \frac{1}{4}x - \frac{3}{10} \) → slope \( m = \frac{1}{4} = 0.25 \)
4. \( y = 1.3x - 1 \) → slope \( m = 1.3 \)
5. \( y = 0.3x + 1 \) → slope \( m = 0.3 \)
6. \( y = \frac{8}{5}x - 1 \) → slope \( m = \frac{8}{5} = 1.6 \)
7. \( y = \frac{3}{10}x \) → slope \( m = \frac{3}{10} = 0.3 \)
8. \( y = 1.6x + 7 \) → slope \( m = 1.6 \)
9. \( y = \frac{3}{2}x - 6 \) → slope \( m = \frac{3}{2} = 1.5 \)

#### Step 2: Group equations by slope
- Slope \( m = 1.5 \): \( y = 1.5x + 3 \), \( y = \frac{3}{2}x - 6 \)
- Slope \( m = 0.25 \): \( y = 0.25x \), \( y = \frac{1}{4}x - \frac{3}{10} \)
- Slope \( m = 0.3 \): \( y = 0.3x + 1 \), \( y = \frac{3}{10}x \)
- Slope \( m = 1.6 \): \( y = \frac{8}{5}x - 1 \), \( y = 1.6x + 7 \)
- Slope \( m = 1.3 \): \( y = 1.3x - 1 \)

#### Step 3: Identify the leftover equation
The equation \( y = 1.3x - 1 \) does not have a match. The leftover equation is:
\[ \boxed{y = 1.3x - 1} \]

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Grid 4

The equations are:
1. \( y = 0.3x + 2 \)
2. \( y = \frac{1}{9}x - 8 \)
3. \( y = \frac{1}{7}x - 6 \)
4. \( y = \frac{1}{3}x - 6 \)
5. \( y = 9x + 1 \)
6. \( y = \frac{1}{6}x \)
7. \( y = 0.1x + 9 \)
8. \( 7y = x - 42 \)
9. \( 6y = x \)

#### Step 1: Identify the slopes
1. \( y = 0.3x + 2 \) → slope \( m = 0.3 \)
2. \( y = \frac{1}{9}x - 8 \) → slope \( m = \frac{1}{9} \)
3. \( y = \frac{1}{7}x - 6 \) → slope \( m = \frac{1}{7} \)
4. \( y = \frac{1}{3}x - 6 \) → slope \( m = \frac{1}{3} \)
5. \( y = 9x + 1 \) → slope \( m = 9 \)
6. \( y = \frac{1}{6}x \) → slope \( m = \frac{1}{6} \)
7. \( y = 0.1x + 9 \) → slope \( m = 0.1 \)
8. \( 7y = x - 42 \) → divide by 7: \( y = \frac{1}{7}x - 6 \) → slope \( m = \frac{1}{7} \)
9. \( 6y = x \) → divide by 6: \( y = \frac{1}{6}x \) → slope \( m = \frac{1}{6} \)

#### Step 2: Group equations by slope
- Slope \( m = \frac{1}{7} \): \( y = \frac{1}{7}x - 6 \), \( 7y = x - 42 \)
- Slope \( m = \frac{1}{6} \): \( y = \frac{1}{6}x \), \( 6y = x \)
- Slope \( m = \frac{1}{9} \): \( y = \frac{1}{9}x - 8 \)
- Slope \( m = \frac{1}{3} \): \( y = \frac{1}{3}x - 6 \)
- Slope \( m = 0.3 \): \( y = 0.3x + 2 \)
- Slope \( m = 9 \): \( y = 9x + 1 \)
- Slope \( m = 0.1 \): \( y = 0.1x + 9 \)

#### Step 3: Identify the leftover equation
The equation \( y = \frac{1}{9}x - 8 \) does not have a match. The leftover equation is:
\[ \boxed{y = \frac{1}{9}x - 8} \]

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Final Answer

The leftover equations from each grid are:
1. \( \boxed{y = 3x + 3} \)
2. \( \boxed{y = 5} \)
3. \( \boxed{y = 1.3x - 1} \)
4. \( \boxed{y = \frac{1}{9}x - 8} \)