Interpretation:

• Slope tells you: "For every 1 unit you move right, how much do you move up or down?"
• m = 2 means: "up 2 for every 1 right"
• m = -3 means: "down 3 for every 1 right"
• m = 1/2 means: "up 1 for every 2 right"

Types of Slope:

Positive Slope (m > 0): Line rises from left to right Example: m = 3, the line goes upward

Negative Slope (m < 0): Line falls from left to right Example: m = -2, the line goes downward

Zero Slope (m = 0): Horizontal line Example: y = 5 (no x term, so m = 0)

Undefined Slope: Vertical line (cannot be written in slope-intercept form) Example: x = 3

Understanding Y-Intercept (b)

The y-intercept is the point where the line crosses the y-axis.

Key facts:

• At the y-intercept, x = 0
• Written as the point (0, b)
• Represents the "starting value" in many real-world problems

Examples:

• y = 2x + 5 has y-intercept (0, 5)
• y = -x - 3 has y-intercept (0, -3)
• y = 4x has y-intercept (0, 0)

Finding Slope from Two Points

Given two points (x₁, y₁) and (x₂, y₂), use:

m = (y₂ - y₁)/(x₂ - x₁)

Example 1: Find slope through (2, 5) and (6, 13)

m = (13 - 5)/(6 - 2) m = 8/4 m = 2

Example 2: Find slope through (-1, 4) and (3, -2)

m = (-2 - 4)/(3 - (-1)) m = -6/4 m = -3/2

Example 3: Find slope through (2, 5) and (7, 5)

m = (5 - 5)/(7 - 2) m = 0/5 m = 0 (horizontal line)

Writing Equations Given Slope and Y-Intercept

This is the easiest case - just substitute into y = mx + b!

Example 1: Write equation with slope 3 and y-intercept -2

m = 3, b = -2 y = 3x - 2

Example 2: Write equation with slope -1/2 and y-intercept 4

m = -1/2, b = 4 y = (-1/2)x + 4

Example 3: Write equation with slope 0 and y-intercept 7

m = 0, b = 7 y = 0x + 7 y = 7 (horizontal line)

Writing Equations Given Slope and One Point

Steps:

1. Use y = mx + b
2. Substitute the slope for m
3. Substitute the point's coordinates for x and y
4. Solve for b
5. Write final equation

Example 1: Write equation with slope 2 passing through (3, 8)

Step 1: y = mx + b Step 2: y = 2x + b Step 3: Substitute (3, 8) 8 = 2(3) + b 8 = 6 + b b = 2 Step 4: y = 2x + 2

Check: Does (3, 8) work? 8 = 2(3) + 2 = 8 ✓

Example 2: Write equation with slope -3 passing through (1, 5)

y = -3x + b 5 = -3(1) + b 5 = -3 + b b = 8

Equation: y = -3x + 8

Example 3: Write equation with slope 1/2 passing through (4, 7)

y = (1/2)x + b 7 = (1/2)(4) + b 7 = 2 + b b = 5

Equation: y = (1/2)x + 5

Writing Equations Given Two Points

Steps:

1. Find the slope using m = (y₂ - y₁)/(x₂ - x₁)
2. Use the slope and one of the points
3. Substitute into y = mx + b
4. Solve for b
5. Write final equation

Example 1: Write equation through (1, 3) and (5, 11)

Step 1: Find slope m = (11 - 3)/(5 - 1) = 8/4 = 2

Step 2: Use m = 2 and point (1, 3) 3 = 2(1) + b 3 = 2 + b b = 1

Equation: y = 2x + 1

Check with other point (5, 11): 11 = 2(5) + 1 = 11 ✓

Example 2: Write equation through (-2, 7) and (3, -3)

Step 1: Find slope m = (-3 - 7)/(3 - (-2)) = -10/5 = -2

Step 2: Use m = -2 and point (3, -3) -3 = -2(3) + b -3 = -6 + b b = 3

Equation: y = -2x + 3

Example 3: Write equation through (0, 4) and (2, 10)

Step 1: Find slope m = (10 - 4)/(2 - 0) = 6/2 = 3

Step 2: Notice (0, 4) is the y-intercept! So b = 4 immediately

Equation: y = 3x + 4

Writing Equations from a Graph

Method 1: Identify slope and y-intercept directly

• Find where line crosses y-axis (that's b)
• Count rise/run between two clear points (that's m)
• Write y = mx + b

Method 2: Use two points from the graph

• Identify coordinates of two points
• Calculate slope
• Find y-intercept
• Write equation

Example: Graph shows line through (0, 2) and (3, 8)

Y-intercept: (0, 2), so b = 2 Slope: m = (8 - 2)/(3 - 0) = 6/3 = 2

Equation: y = 2x + 2

Parallel Lines

Parallel lines have the SAME slope but different y-intercepts.

Example 1: Write equation parallel to y = 3x + 1 through (2, 10)

Parallel means same slope: m = 3 Using point (2, 10): 10 = 3(2) + b 10 = 6 + b b = 4

Equation: y = 3x + 4

Example 2: Write equation parallel to y = -2x - 5 with y-intercept 3

Same slope: m = -2 Given y-intercept: b = 3

Equation: y = -2x + 3

Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals.

If line 1 has slope m₁, perpendicular line has slope m₂ = -1/m₁

Key relationship: m₁ × m₂ = -1

Example 1: Write equation perpendicular to y = 2x + 1 through (4, 5)

Original slope: m₁ = 2 Perpendicular slope: m₂ = -1/2

Using point (4, 5): 5 = (-1/2)(4) + b 5 = -2 + b b = 7

Equation: y = (-1/2)x + 7

Example 2: Write equation perpendicular to y = -3x + 2 through (6, 1)

Original slope: m₁ = -3 Perpendicular slope: m₂ = -1/(-3) = 1/3

Using point (6, 1): 1 = (1/3)(6) + b 1 = 2 + b b = -1

Equation: y = (1/3)x - 1

Converting to Slope-Intercept Form

If an equation is in standard form (Ax + By = C), solve for y to get slope-intercept form.

Example 1: Convert 2x + 3y = 12 to slope-intercept form

3y = -2x + 12 y = (-2/3)x + 4

Slope: m = -2/3, y-intercept: b = 4

Example 2: Convert 5x - 2y = 10 to slope-intercept form

-2y = -5x + 10 y = (5/2)x - 5

Example 3: Convert x + y = 7 to slope-intercept form

y = -x + 7

Slope: m = -1, y-intercept: b = 7

Real-World Applications

Slope-intercept form naturally fits many real situations:

Example 1: Phone Plan A phone plan costs $30/month plus$0.10 per text message.

Let x = number of texts Let y = total cost

Equation: y = 0.10x + 30

• Slope (0.10): cost per text
• Y-intercept (30): base monthly fee

Example 2: Temperature Conversion Fahrenheit to Celsius: C = (5/9)(F - 32)

Rearranged: C = (5/9)F - 160/9

• Slope (5/9): rate of change
• Y-intercept (-160/9): offset

Example 3: Car Rental Rent a car for $40 plus$0.25 per mile driven.

Equation: y = 0.25x + 40

• x: miles driven
• y: total cost
• Slope: $0.25 per mile
• Y-intercept: $40 base fee

Example 4: Water Draining A pool with 1000 gallons drains at 50 gallons per hour.

Equation: y = -50x + 1000

• x: hours
• y: gallons remaining
• Slope: -50 (negative = decreasing)
• Y-intercept: 1000 (starting amount)

Common Mistakes to Avoid

1. Confusing m and b In y = 3x + 5: slope is 3, not 5!

2. Sign errors with negative slopes y = -2x + 3, not y = 2x - 3

3. Not simplifying fractions Write y = (2/4)x + 1 as y = (1/2)x + 1

4. Wrong order in slope formula m = (y₂ - y₁)/(x₂ - x₁), be consistent!

5. Forgetting negative reciprocal for perpendicular Perpendicular to m = 2 is m = -1/2, not m = 1/2

Problem-Solving Strategy

Given slope and y-intercept: Direct substitution Given slope and point: Substitute and solve for b Given two points: Find slope first, then find b From a graph: Identify points and calculate Parallel lines: Same slope, different b Perpendicular lines: Negative reciprocal slope

Quick Reference

Practice Tips

• Always identify what you're given (slope? points? graph?)
• Write y = mx + b and fill in what you know
• Show all substitution steps
• Check your equation with given points
• Practice converting between forms
• Memorize: parallel = same slope, perpendicular = negative reciprocal
• In word problems, identify what slope and y-intercept represent

Substitute: $y = 3x + (-2)$ $y = 3x - 2$

Answer: $y = 3x - 2$