Math worksheet for identifying parallel, perpendicular, or neither relationships between linear equations.

Let's solve this step-by-step. The task is to determine whether the two lines in each box are parallel, perpendicular, or neither.

🔷 Key Concepts:

- Parallel lines: Have the same slope (m₁ = m₂)
- Perpendicular lines: Have slopes that are negative reciprocals of each other (m₁ × m₂ = -1)
- Neither: If neither of the above conditions is met

We will analyze each pair of equations and find their slopes.

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✔ Step 1: Convert equations to slope-intercept form:

$$ y = mx + b $$
where $ m $ is the slope.

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## 🟨 Box 1:
$$
y = 3x - 7 \\
y = 3x + 1
$$
- Both have slope $ m = 3 $
- Same slope → Parallel

✔ Color: Yellow

---

## 🟦 Box 2:
$$
y = -\frac{2}{5}x + 3 \\
y = \frac{2}{5}x + 8
$$
- Slopes: $ -\frac{2}{5} $ and $ \frac{2}{5} $
- Not equal → not parallel
- Product: $ (-\frac{2}{5})(\frac{2}{5}) = -\frac{4}{25} \ne -1 $ → not perpendicular

✔ Neither

→ Uncolored

---

## 🟦 Box 3:
$$
y = -\frac{1}{4}x \\
y = 4x - 5
$$
- Slopes: $ -\frac{1}{4} $ and $ 4 $
- Check product: $ (-\frac{1}{4})(4) = -1 $ → negative reciprocal → Perpendicular

✔ Color: Light Blue

---

## 🟨 Box 4:
$$
2x + 7y = 28 \\
7x - 2y = 4
$$
Convert both:

1. $ 2x + 7y = 28 $ → $ 7y = -2x + 28 $ → $ y = -\frac{2}{7}x + 4 $ → $ m_1 = -\frac{2}{7} $

2. $ 7x - 2y = 4 $ → $ -2y = -7x + 4 $ → $ y = \frac{7}{2}x - 2 $ → $ m_2 = \frac{7}{2} $

Check product: $ (-\frac{2}{7})(\frac{7}{2}) = -1 $ → Perpendicular

✔ Color: Light Blue

---

## 🟦 Box 5:
$$
y = -5x + 1 \\
x - 5y = 30
$$
Second equation: $ x - 5y = 30 $ → $ -5y = -x + 30 $ → $ y = \frac{1}{5}x - 6 $ → $ m_2 = \frac{1}{5} $

First slope: $ m_1 = -5 $

Product: $ (-5)(\frac{1}{5}) = -1 $ → Perpendicular

✔ Color: Light Blue

---

## 🟨 Box 6:
$$
3x + 2y = 8 \\
2x + 3y = -12
$$
Convert:

1. $ 3x + 2y = 8 $ → $ 2y = -3x + 8 $ → $ y = -\frac{3}{2}x + 4 $ → $ m_1 = -\frac{3}{2} $

2. $ 2x + 3y = -12 $ → $ 3y = -2x -12 $ → $ y = -\frac{2}{3}x - 4 $ → $ m_2 = -\frac{2}{3} $

Slopes: $ -\frac{3}{2} $ vs $ -\frac{2}{3} $ — not equal, not negative reciprocals
$ (-\frac{3}{2})(-\frac{2}{3}) = 1 \ne -1 $ → Neither

✔ Uncolored

---

## 🟦 Box 7:
$$
y = -4x - 1 \\
8x + 2y = 14
$$
Second: $ 8x + 2y = 14 $ → $ 2y = -8x + 14 $ → $ y = -4x + 7 $ → $ m_2 = -4 $

First: $ m_1 = -4 $

Same slope → Parallel

✔ Color: Yellow

---

## 🟦 Box 8:
$$
x + y = 7 \\
x - y = 9
$$
Convert:

1. $ x + y = 7 $ → $ y = -x + 7 $ → $ m_1 = -1 $

2. $ x - y = 9 $ → $ -y = -x + 9 $ → $ y = x - 9 $ → $ m_2 = 1 $

Product: $ (-1)(1) = -1 $ → Perpendicular

✔ Color: Light Blue

---

## 🟦 Box 9:
$$
y = \frac{1}{3}x + 9 \\
x - 3y = 3
$$
Second: $ x - 3y = 3 $ → $ -3y = -x + 3 $ → $ y = \frac{1}{3}x - 1 $ → $ m_2 = \frac{1}{3} $

First: $ m_1 = \frac{1}{3} $

Same slope → Parallel

✔ Color: Yellow

---

## 🟦 Box 10:
$$
4x + 9y = 18 \\
y = 4x + 9
$$
First: $ 4x + 9y = 18 $ → $ 9y = -4x + 18 $ → $ y = -\frac{4}{9}x + 2 $ → $ m_1 = -\frac{4}{9} $

Second: $ m_2 = 4 $

Not same → not parallel
Product: $ (-\frac{4}{9})(4) = -\frac{16}{9} \ne -1 $ → not perpendicular

✔ Neither → Uncolored

---

## 🟦 Box 11:
$$
5x - 10y = 20 \\
y = -2x + 6
$$
First: $ 5x - 10y = 20 $ → $ -10y = -5x + 20 $ → $ y = \frac{1}{2}x - 2 $ → $ m_1 = \frac{1}{2} $

Second: $ m_2 = -2 $

Product: $ (\frac{1}{2})(-2) = -1 $ → Perpendicular

✔ Color: Light Blue

---

## 🟦 Box 12:
$$
-9x + 12y = 24 \\
y = \frac{3}{4}x - 5
$$
First: $ -9x + 12y = 24 $ → $ 12y = 9x + 24 $ → $ y = \frac{3}{4}x + 2 $ → $ m_1 = \frac{3}{4} $

Second: $ m_2 = \frac{3}{4} $

Same slope → Parallel

✔ Color: Yellow

---

## 🟦 Box 13:
$$
y = x - 3 \\
x - y = 8
$$
Second: $ x - y = 8 $ → $ -y = -x + 8 $ → $ y = x - 8 $ → $ m_2 = 1 $

First: $ m_1 = 1 $

Same slope → Parallel

✔ Color: Yellow

---

## 🟦 Box 14:
$$
10x + 8y = 16 \\
5y = 4x - 15
$$
First: $ 10x + 8y = 16 $ → $ 8y = -10x + 16 $ → $ y = -\frac{5}{4}x + 2 $ → $ m_1 = -\frac{5}{4} $

Second: $ 5y = 4x - 15 $ → $ y = \frac{4}{5}x - 3 $ → $ m_2 = \frac{4}{5} $

Product: $ (-\frac{5}{4})(\frac{4}{5}) = -1 $ → Perpendicular

✔ Color: Light Blue

---

## 🟦 Box 15:
$$
y = \frac{5}{3}x + 7 \\
6x - 10y = 10
$$
Second: $ 6x - 10y = 10 $ → $ -10y = -6x + 10 $ → $ y = \frac{3}{5}x - 1 $ → $ m_2 = \frac{3}{5} $

First: $ m_1 = \frac{5}{3} $

Check product: $ (\frac{5}{3})(\frac{3}{5}) = 1 \ne -1 $ → not perpendicular
Not same slope → not parallel

✔ Neither → Uncolored

---

## 🟦 Box 16:
$$
x - 2y = 18 \\
2x + y = 6
$$
Convert:

1. $ x - 2y = 18 $ → $ -2y = -x + 18 $ → $ y = \frac{1}{2}x - 9 $ → $ m_1 = \frac{1}{2} $

2. $ 2x + y = 6 $ → $ y = -2x + 6 $ → $ m_2 = -2 $

Product: $ (\frac{1}{2})(-2) = -1 $ → Perpendicular

✔ Color: Light Blue

---

## 🟦 Box 17:
$$
x = 4 \\
x = -6
$$
Both are vertical lines (constant x)

Vertical lines are parallel to each other (same direction, never intersect)

✔ Color: Yellow

---

## 🟦 Box 18:
$$
x = 1 \\
y = -8
$$
- $ x = 1 $: vertical line
- $ y = -8 $: horizontal line

A vertical and a horizontal line are perpendicular

✔ Color: Light Blue

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✔ Final Answer Summary (by box):

| Box | Relationship | Color |
|-----|--------------|--------|
| 1 | Parallel | Yellow |
| 2 | Neither | Uncolored |
| 3 | Perpendicular| Light Blue |
| 4 | Perpendicular| Light Blue |
| 5 | Perpendicular| Light Blue |
| 6 | Neither | Uncolored |
| 7 | Parallel | Yellow |
| 8 | Perpendicular| Light Blue |
| 9 | Parallel | Yellow |
| 10 | Neither | Uncolored |
| 11 | Perpendicular| Light Blue |
| 12 | Parallel | Yellow |
| 13 | Parallel | Yellow |
| 14 | Perpendicular| Light Blue |
| 15 | Neither | Uncolored |
| 16 | Perpendicular| Light Blue |
| 17 | Parallel | Yellow |
| 18 | Perpendicular| Light Blue |

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🎯 Final Coloring Guide:

- Yellow (Parallel): Boxes 1, 7, 9, 12, 13, 17
- Light Blue (Perpendicular): Boxes 3, 4, 5, 8, 11, 14, 16, 18
- Uncolored (Neither): Boxes 2, 6, 10, 15

Let me know if you'd like a visual version or printable answer key!