Writing Linear Equations

Slope-intercept form and point-slope form

Writing Linear Equations (Slope-Intercept Form)

What is Slope-Intercept Form?

Slope-intercept form is the most common way to write linear equations:

y = mx + b

Where:

m is the slope (rate of change)
b is the y-intercept (where the line crosses the y-axis)
x and y are variables

This form is called "slope-intercept" because it directly shows the slope and y-intercept!

Why Use This Form?

Slope-intercept form is useful because:

You can immediately identify the slope and y-intercept
It's easy to graph
It clearly shows the rate of change and starting value
It's perfect for real-world applications

Understanding Slope (m)

Slope measures how steep a line is and its direction.

Formula: m = rise/run = (change in y)/(change in x) = (y₂ - y₁)/(x₂ - x₁)

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:

Use y = mx + b
Substitute the slope for m
Substitute the point's coordinates for x and y
Solve for b
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:

Find the slope using m = (y₂ - y₁)/(x₂ - x₁)
Use the slope and one of the points
Substitute into y = mx + b
Solve for b
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

Confusing m and b In y = 3x + 5: slope is 3, not 5!
Sign errors with negative slopes y = -2x + 3, not y = 2x - 3
Not simplifying fractions Write y = (2/4)x + 1 as y = (1/2)x + 1
Wrong order in slope formula m = (y₂ - y₁)/(x₂ - x₁), be consistent!
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

| Given Information | Strategy | |-------------------|----------| | m and b | Write y = mx + b directly | | m and (x₁, y₁) | Substitute point, solve for b | | (x₁, y₁) and (x₂, y₂) | Find m, then find b | | Graph | Identify b, calculate m | | Parallel to y = mx + b | Use same m, find new b | | Perpendicular to y = mx + b | Use m_perp = -1/m, find b |

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

📚 Practice Problems

1Problem 1easy

❓ Question:

What is the slope and y-intercept of y = 3x - 5?

💡 Show Solution

Step 1: Identify the slope-intercept form: y = mx + b where m = slope and b = y-intercept

Step 2: Compare to the given equation: y = 3x - 5

Step 3: Identify m (slope): m = 3

Step 4: Identify b (y-intercept): b = -5

Step 5: Interpret: Slope of 3 means rise/run = 3/1 The line goes up 3 units for every 1 unit right Y-intercept of -5 means the line crosses y-axis at (0, -5)

Answer: Slope = 3, y-intercept = -5

2Problem 2easy

❓ Question:

Write an equation for a line with slope 3 and y-intercept -2

💡 Show Solution

Use slope-intercept form: $y = mx + b$

Given: $m = 3$ and $b = -2$

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

Answer: $y = 3x - 2$

3Problem 3easy

❓ Question:

Write the equation of a line with slope -2 and y-intercept 7.

💡 Show Solution

Step 1: Recall slope-intercept form: y = mx + b

Step 2: Substitute given values: m = -2 (slope) b = 7 (y-intercept)

Step 3: Write the equation: y = -2x + 7

Step 4: Verify it makes sense: When x = 0: y = -2(0) + 7 = 7 ✓ (passes through (0, 7)) Slope is negative, so line goes down from left to right ✓

Answer: y = -2x + 7

4Problem 4medium

❓ Question:

Write an equation for the line passing through $(2, 5)$ with slope $-4$

💡 Show Solution

Use point-slope form: $y - y_1 = m(x - x_1)$

Given: $(x_1, y_1) = (2, 5)$ and $m = -4$

Substitute: $y - 5 = -4(x - 2)$

Expand: $y - 5 = -4x + 8$

Add 5 to both sides: $y = -4x + 13$

Answer: $y = -4x + 13$

5Problem 5medium

❓ Question:

Convert 2x + 3y = 12 to slope-intercept form.

💡 Show Solution

Step 1: Goal - isolate y on one side: 2x + 3y = 12

Step 2: Subtract 2x from both sides: 3y = -2x + 12

Step 3: Divide everything by 3: y = -2x/3 + 12/3 y = -2/3 x + 4

Step 4: Identify slope and y-intercept: Slope: m = -2/3 Y-intercept: b = 4

Step 5: Verify: Check with original equation at (0, 4): 2(0) + 3(4) = 0 + 12 = 12 ✓

Answer: y = -2/3 x + 4

6Problem 6medium

❓ Question:

Write the equation of a line that passes through (0, -3) with slope 4.

💡 Show Solution

Step 1: Identify what we know: The point (0, -3) is the y-intercept (where x = 0) So b = -3 Slope m = 4

Step 2: Use slope-intercept form: y = mx + b

Step 3: Substitute values: y = 4x + (-3) y = 4x - 3

Step 4: Verify with the given point: When x = 0: y = 4(0) - 3 = -3 ✓ Point (0, -3) is on the line ✓

Step 5: Check the slope with another point: When x = 1: y = 4(1) - 3 = 1 From (0, -3) to (1, 1): Slope = (1 - (-3))/(1 - 0) = 4/1 = 4 ✓

Answer: y = 4x - 3

7Problem 7hard

❓ Question:

Write an equation for the line passing through $(1, 3)$ and $(4, 9)$

💡 Show Solution

Step 1: Find the slope $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$

Step 2: Use point-slope form with $(1, 3)$ $y - 3 = 2(x - 1)$

Step 3: Simplify to slope-intercept form $y - 3 = 2x - 2$ $y = 2x + 1$

Answer: $y = 2x + 1$

8Problem 8hard

❓ Question:

Write the equation in slope-intercept form of the line passing through (2, 5) and (6, 13).

💡 Show Solution

Step 1: Find the slope using two points: m = (y₂ - y₁)/(x₂ - x₁) m = (13 - 5)/(6 - 2) m = 8/4 m = 2

Step 2: Use point-slope form first: y - y₁ = m(x - x₁) Using point (2, 5): y - 5 = 2(x - 2)

Step 3: Solve for y (convert to slope-intercept form): y - 5 = 2x - 4 y = 2x - 4 + 5 y = 2x + 1

Step 4: Verify with both points: Point (2, 5): y = 2(2) + 1 = 5 ✓ Point (6, 13): y = 2(6) + 1 = 13 ✓

Step 5: Alternative method - use y = mx + b directly: We know m = 2, so y = 2x + b Use point (2, 5): 5 = 2(2) + b 5 = 4 + b b = 1 Therefore: y = 2x + 1 ✓

Answer: y = 2x + 1

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