Writing Linear Equations

Review: Forms of Linear Equations

Slope-intercept form: y = mx + b

• m = slope, b = y-intercept

Point-slope form: y - y₁ = m(x - x₁)

• m = slope, (x₁, y₁) = known point

Standard form: Ax + By = C

• A, B, C are integers; A ≥ 0

Writing from Slope and Y-Intercept

Given: Slope (m) and y-intercept (b)

Use: y = mx + b (directly!)

Example 1: Basic

Slope = 3, y-intercept = -2

Write: y = 3x - 2

That's it!

Example 2: Negative Slope

Slope = -1/2, y-intercept = 4

Write: y = (-1/2)x + 4 or y = -x/2 + 4

Writing from Slope and One Point

Given: Slope (m) and one point (x₁, y₁)

Method 1: Point-slope form

Formula: y - y₁ = m(x - x₁)

Example 1: Using Point-Slope

Slope = 2, point (3, 5)

Step 1: Substitute into point-slope form

y - 5 = 2(x - 3)

Step 2: Simplify to slope-intercept (optional)

y - 5 = 2x - 6
y = 2x - 1

Answer: y = 2x - 1

Example 2: Negative Slope

Slope = -3, point (-2, 4)

Point-slope:

y - 4 = -3(x - (-2))
y - 4 = -3(x + 2)
y - 4 = -3x - 6
y = -3x - 2

Answer: y = -3x - 2

Writing from Two Points

Given: Two points (x₁, y₁) and (x₂, y₂)

Steps:

1. Find slope: m = (y₂ - y₁)/(x₂ - x₁)
2. Use point-slope with either point
3. Simplify to slope-intercept

Example 1: Two Points

Points: (2, 5) and (6, 13)

Step 1: Find slope

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

Step 2: Use point-slope with (2, 5)

y - 5 = 2(x - 2)
y - 5 = 2x - 4
y = 2x + 1

Answer: y = 2x + 1

Check with (6, 13): 13 = 2(6) + 1 = 13 ✓

Example 2: Negative Slope

Points: (1, 7) and (4, -2)

Find slope:

m = (-2 - 7)/(4 - 1)
m = -9/3
m = -3

Use (1, 7):

y - 7 = -3(x - 1)
y - 7 = -3x + 3
y = -3x + 10

Answer: y = -3x + 10

Example 3: With Fractions

Points: (0, 3) and (6, 5)

Find slope:

m = (5 - 3)/(6 - 0)
m = 2/6
m = 1/3

Use (0, 3): (This is the y-intercept!)

y = (1/3)x + 3

Answer: y = (1/3)x + 3

Horizontal and Vertical Lines

Horizontal line: y = k

• Slope = 0
• All points have same y-coordinate

Vertical line: x = c

• Slope = undefined
• All points have same x-coordinate
• NOT a function!

Example: Horizontal Line

Through point (5, -2)

All points have y = -2

Equation: y = -2

Example: Vertical Line

Through point (3, 7)

All points have x = 3

Equation: x = 3

Parallel Lines

Parallel lines: Same slope, different y-intercepts

If line has slope m, parallel line also has slope m

Example: Write Parallel Line

Write equation parallel to y = 2x + 1 through (3, 5)

Step 1: Identify slope of given line

• m = 2

Step 2: Use same slope with new point

y - 5 = 2(x - 3)
y - 5 = 2x - 6
y = 2x - 1

Answer: y = 2x - 1

Perpendicular Lines

Perpendicular lines: Slopes are negative reciprocals

If one slope is m, perpendicular slope is -1/m

Examples:

• 2 and -1/2
• -3 and 1/3
• 1/4 and -4

Example: Write Perpendicular Line

Write equation perpendicular to y = (2/3)x + 4 through (6, 1)

Step 1: Find perpendicular slope

• Original slope: 2/3
• Perpendicular: -3/2

Step 2: Use with point (6, 1)

y - 1 = (-3/2)(x - 6)
y - 1 = (-3/2)x + 9
y = (-3/2)x + 10

Answer: y = (-3/2)x + 10

Writing from Word Problems

Translate words to math!

Example 1: Rental Cost

Problem: Rent a truck for $30 plus $0.50 per mile. Write equation for total cost.**

Let:

• x = miles driven
• y = total cost

Fixed cost: $30 (y-intercept) Rate: $0.50 per mile (slope)

Equation: y = 0.50x + 30

Example 2: Temperature Conversion

Problem: Water freezes at 0°C (32°F). Water boils at 100°C (212°F). Write equation.**

Points: (0, 32) and (100, 212)

Find slope:

m = (212 - 32)/(100 - 0)
m = 180/100
m = 9/5

Use (0, 32):

F = (9/5)C + 32

This is the temperature conversion formula!

Example 3: Savings Account

Problem: Account has $500. Save $75 per month. Write equation.**

Initial: $500 (y-intercept) Rate: $75 per month (slope)

Equation: y = 75x + 500

• x = months
• y = account balance

Converting Forms

From slope-intercept to standard:

Example: Convert to Standard Form

Given: y = (2/3)x - 4

Clear fractions: Multiply by 3

3y = 2x - 12

Rearrange: Ax + By = C

-2x + 3y = -12

Make A positive:

2x - 3y = 12

Standard form: 2x - 3y = 12

Practice