Drag the tiles to the correct boxes to complete the pairs.

Determine whether each pair of lines is perpendicular, parallel, or neither.

[tex]\[
\begin{array}{lll}
y = 2x + 4 & 2y = 4x + 4 & 4y = 2x - 4 \\
2y = 4x - 7 & y = -2x - 2 & y = -2x + 9 \\
\end{array}
\][/tex]

Neither [tex]$\square$[/tex] \\
Parallel [tex]$\square$[/tex] \\
Perpendicular [tex]$\square$[/tex]



Answer :

To solve this problem, we need to determine the slopes of each pair of lines and then compare them. Recall that:

- Slopes of parallel lines are equal.
- Slopes of perpendicular lines are negative reciprocals of each other.

Given equations:

1. [tex]\( y = 2x + 4 \)[/tex]
2. [tex]\( 2y = 4x + 4 \)[/tex]
3. [tex]\( 4y = 2x - 4 \)[/tex]
4. [tex]\( 2y = 4x - 7 \)[/tex]
5. [tex]\( y = -2x - 2 \)[/tex]
6. [tex]\( y = -2x + 9 \)[/tex]

First, simplify each equation to the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.

1. [tex]\( y = 2x + 4 \)[/tex]
- Slope [tex]\( m = 2 \)[/tex]

2. [tex]\( 2y = 4x + 4 \)[/tex]
- Divide both sides by 2: [tex]\( y = 2x + 2 \)[/tex]
- Slope [tex]\( m = 2 \)[/tex]

3. [tex]\( 4y = 2x - 4 \)[/tex]
- Divide both sides by 4: [tex]\( y = \frac{1}{2}x - 1 \)[/tex]
- Slope [tex]\( m = \frac{1}{2} \)[/tex]

4. [tex]\( 2y = 4x - 7 \)[/tex]
- Divide both sides by 2: [tex]\( y = 2x - \frac{7}{2} \)[/tex]
- Slope [tex]\( m = 2 \)[/tex]

5. [tex]\( y = -2x - 2 \)[/tex]
- Slope [tex]\( m = -2 \)[/tex]

6. [tex]\( y = -2x + 9 \)[/tex]
- Slope [tex]\( m = -2 \)[/tex]

Next, compare the slopes:

- Lines 1, 2, and 4 all have the slope [tex]\( m = 2 \)[/tex], making them parallel to each other.
- Lines 5 and 6 both have the slope [tex]\( m = -2 \)[/tex], making them parallel to each other.
- No lines have slopes that are negative reciprocals, so none are perpendicular.
- Lines 3 and any of the other lines (1, 2, 4, 5, 6) have different slopes, so they are neither parallel nor perpendicular to each other.

Thus:

Neither:

[tex]\[ \boxed{4y = 2x - 4 \text{ and any of the other lines (e.g., line 1)}} \][/tex]

Parallel:

[tex]\[ \text{Box 1: } y = 2x + 4 \text{ and } 2y = 4x + 4 \][/tex]

[tex]\[ \text{Box 2: } y = -2x - 2 \text{ and } y = -2x + 9 \][/tex]

No lines are perpendicular in the given set.