Linear Relationships

What is a Linear Relationship?

A linear relationship has a constant rate of change between two variables.

Key feature: When graphed, makes a straight line!

Examples:

• Distance = speed × time (constant speed)
• Total cost = price per item × number of items

Rate of Change

Rate of change tells how much one variable changes when the other changes.

Formula: Rate of Change = Change in y / Change in x

Also called: Slope!

Example 1: Constant Rate

Change: Every +1 hour → +$15 earnings

Rate of change: $15 per hour (constant!)

Linear? YES! Makes a straight line.

Example 2: NOT Constant

Change:

• 1 to 2: +3
• 2 to 3: +5
• 3 to 4: +7

Rate of change: NOT constant!

Linear? NO! (This is y = x², a parabola)

Finding Rate of Change from Table

Pick any two points:

Rate = (y₂ − y₁) / (x₂ − x₁)

Example: Calculate Rate

Use points (2,7) and (6,19):

Rate = (19 − 7) / (6 − 2) = 12 / 4 = 3

Answer: Rate of change is 3

Check with another pair (4,13) and (6,19):

Rate = (19 − 13) / (6 − 4) = 6 / 2 = 3 ✓

Constant rate → Linear!

Understanding Slope

Slope (m) is the rate of change in a linear relationship.

Slope = rise / run

Where:

• Rise: vertical change (up/down)
• Run: horizontal change (left/right)

Example: Positive Slope

Line goes through (1,2) and (3,8)

Rise: 8 − 2 = 6 (up) Run: 3 − 1 = 2 (right)

Slope: m = 6/2 = 3

Meaning: For every 1 unit right, go up 3 units

Example: Negative Slope

Line goes through (0,10) and (5,0)

Rise: 0 − 10 = −10 (down) Run: 5 − 0 = 5 (right)

Slope: m = −10/5 = −2

Meaning: For every 1 unit right, go down 2 units

Types of Slope

Positive slope (m > 0): Line goes up ↗

• As x increases, y increases

Negative slope (m < 0): Line goes down ↘

• As x increases, y decreases

Zero slope (m = 0): Horizontal line →

• y stays constant

Undefined slope: Vertical line ↕

• x stays constant
• NOT a function!

Proportional Relationships

A proportional relationship is a special linear relationship that passes through the origin (0,0).

Formula: y = kx

Where k is the constant of proportionality (the slope)

No y-intercept (or y-intercept = 0)

Example: Proportional

Relationship: y = 3x

Proportional? YES! (Goes through origin)

Constant of proportionality: k = 3

Example: Linear but NOT Proportional

Relationship: y = 3x + 2

Linear? YES! (Constant rate of change = 3)

Proportional? NO! (Doesn't go through origin)

Direct Variation

Direct variation is another name for proportional relationship.

Format: y varies directly with x Equation: y = kx

Example: Direct Variation

"y varies directly with x, and y = 12 when x = 3"

Find k:

y = kx
12 = k(3)
k = 4

Equation: y = 4x

Find y when x = 7:

y = 4(7) = 28

Identifying Linear Relationships

From a table:

• Check if rate of change is constant
• Differences in y should be equal for equal differences in x

From a graph:

• Must be a straight line

From an equation:

• Highest power of x is 1
• y = mx + b (YES)
• y = x² (NO)

Real-World Linear Relationships

Car rental: $50 + $20 per day

• y = 20x + 50
• Rate of change: $20 per day

Phone plan: $30 base + $0.10 per minute

• y = 0.10x + 30
• Rate of change: $0.10 per minute

Distance: 60 mph for x hours

• d = 60x
• Proportional relationship!

Temperature conversion: F = (9/5)C + 32

• Linear but not proportional

Comparing Linear Relationships

Example: Two Jobs

Job A: y = 15x (earns $15/hour) Job B: y = 12x + 20 (earns $12/hour + $20 bonus)

Compare:

• Job A: Steeper slope (higher hourly rate)
• Job B: Higher y-intercept (starts with bonus)

When is A better?

• 15x > 12x + 20
• 3x > 20
• x > 6.67
• After about 7 hours, Job A pays more!

Practice