I need help finding the slope and determining wether its parallel ... - Free Printable

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'll go through each box, find the slopes of the two lines, and compare them.

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🟨 Box 1:

Lines:
1. $ y = 3x - 7 $
2. $ y = 3x + 1 $

✔ Both have slope $ m = 3 $ → Same slope

➡️ Parallel → Color: Yellow

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🟦 Box 2:

Lines:
1. $ y = -\frac{2}{5}x + 3 $
2. $ y = \frac{2}{5}x + 8 $

Slopes: $ m_1 = -\frac{2}{5},\ m_2 = \frac{2}{5} $

- Not equal → not parallel
- Product: $ (-\frac{2}{5})(\frac{2}{5}) = -\frac{4}{25} \neq -1 $ → not perpendicular

➡️ Neither → Leave uncolored

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🟨 Box 3:

Lines:
1. $ y = -\frac{1}{4}x $
2. $ y = 4x - 5 $

Slopes: $ m_1 = -\frac{1}{4},\ m_2 = 4 $

Check product: $ (-\frac{1}{4})(4) = -1 $ → Negative reciprocal

➡️ Perpendicular → Color: Light Blue

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🟨 Box 4:

Lines:
1. $ 2x + 7y = 28 $
2. $ 7x - 2y = 4 $

Convert both to slope-intercept form ($ y = mx + b $):

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

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

Product: $ (-\frac{2}{7})(\frac{7}{2}) = -1 $ → Negative reciprocal

➡️ Perpendicular → Color: Light Blue

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🟨 Box 5:

Lines:
1. $ y = -5x + 1 $
2. $ x - 5y = 30 $

Convert second equation:

$ x - 5y = 30 $
→ $ -5y = -x + 30 $
→ $ y = \frac{1}{5}x - 6 $ → $ m_2 = \frac{1}{5} $

First line: $ m_1 = -5 $

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

➡️ Perpendicular → Color: Light Blue

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🟨 Box 6:

Lines:
1. $ 3x + 2y = 8 $
2. $ 2x + 3y = -12 $

Convert both:

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

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

- Slopes different → not parallel
- Product: $ (-\frac{3}{2})(-\frac{2}{3}) = 1 \neq -1 $ → not perpendicular

➡️ Neither → Uncolored

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🟨 Box 7:

Lines:
1. $ y = -4x - 1 $
2. $ 8x + 2y = 14 $

Convert second:

$ 8x + 2y = 14 $ → $ 2y = -8x + 14 $ → $ y = -4x + 7 $ → $ m_2 = -4 $

First line: $ m_1 = -4 $

Same slope → Parallel

➡️ Parallel → Color: Yellow

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🟦 Box 8:

Lines:
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 $ → Negative reciprocal

➡️ Perpendicular → Color: Light Blue

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🟨 Box 9:

Lines:
1. $ y = \frac{1}{3}x + 9 $
2. $ x - 3y = 3 $

Convert second:

$ x - 3y = 3 $ → $ -3y = -x + 3 $ → $ y = \frac{1}{3}x - 1 $ → $ m_2 = \frac{1}{3} $

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

Same slope → Parallel

➡️ Parallel → Color: Yellow

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🟨 Box 10:

Lines:
1. $ 4x + 9y = 18 $
2. $ y = 4x + 9 $

Convert 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} \neq -1 $ → not perpendicular

➡️ Neither → Uncolored

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🟦 Box 11:

Lines:
1. $ 5x - 10y = 20 $
2. $ y = -2x + 6 $

Convert 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 $ → Negative reciprocal

➡️ Perpendicular → Color: Light Blue

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🟨 Box 12:

Lines:
1. $ -9x + 12y = 24 $
2. $ y = \frac{3}{4}x - 5 $

Convert 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

➡️ Parallel → Color: Yellow

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🟦 Box 13:

Lines:
1. $ y = x - 3 $
2. $ x - y = 8 $

Convert second:

$ x - y = 8 $ → $ -y = -x + 8 $ → $ y = x - 8 $ → $ m_2 = 1 $

First: $ m_1 = 1 $

Same slope → Parallel

➡️ Parallel → Color: Yellow

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🟨 Box 14:

Lines:
1. $ 10x + 8y = 16 $
2. $ 5y = 4x - 15 $

Convert both:

Line 1:
$ 10x + 8y = 16 $ → $ 8y = -10x + 16 $ → $ y = -\frac{5}{4}x + 2 $ → $ m_1 = -\frac{5}{4} $

Line 2:
$ 5y = 4x - 15 $ → $ y = \frac{4}{5}x - 3 $ → $ m_2 = \frac{4}{5} $

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

➡️ Perpendicular → Color: Light Blue

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🟨 Box 15:

Lines:
1. $ y = \frac{5}{3}x + 7 $
2. $ 6x - 10y = 10 $

Convert second:

$ 6x - 10y = 10 $ → $ -10y = -6x + 10 $ → $ y = \frac{3}{5}x - 1 $ → $ m_2 = \frac{3}{5} $

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

Product: $ (\frac{5}{3})(\frac{3}{5}) = 1 \neq -1 $ → Not perpendicular

Not same slope → not parallel

➡️ Neither → Uncolored

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🟨 Box 16:

Lines:
1. $ x - 2y = 18 $
2. $ 2x + y = 6 $

Convert both:

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

Line 2:
$ 2x + y = 6 $ → $ y = -2x + 6 $ → $ m_2 = -2 $

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

➡️ Perpendicular → Color: Light Blue

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🟨 Box 17:

Lines:
1. $ x = 4 $
2. $ x = -6 $

These are vertical lines (undefined slope)

Vertical lines are parallel to each other

➡️ Parallel → Color: Yellow

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🟨 Box 18:

Lines:
1. $ x = 1 $
2. $ y = -8 $

- $ x = 1 $: vertical line
- $ y = -8 $: horizontal line

Vertical and horizontal lines are perpendicular

➡️ Perpendicular → Color: Light Blue

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✔ Final Answer Summary:

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

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🎨 Coloring Instructions:

- Yellow: Boxes 1, 7, 9, 12, 13, 17
- Light Blue: Boxes 3, 4, 5, 8, 11, 14, 16, 18
- Uncolored: Boxes 2, 6, 10, 15

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