Writing Linear Equations

Q: How can I study Writing Linear Equations effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 3 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Q: Is this Writing Linear Equations study guide free?

Yes — all study notes, flashcards, and practice problems for Writing Linear Equations on Study Mondo are free to access. No account is needed.

Q: What course covers Writing Linear Equations?

Writing Linear Equations is part of the Algebra 1 course on Study Mondo, specifically in the Linear Equations section. You can explore the full course for more related topics and practice resources.

Q: Are there practice problems for Writing Linear Equations?

Yes, this page includes 3 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Slope-intercept form and point-slope form

Written and reviewed by the Study Mondo Education TeamLast updated April 18, 2026

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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:

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: $y = 3x - 2$

2Problem 2medium

❓ 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: $y = -4x + 13$

3Problem 3hard

❓ 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: $y = 2x + 1$

Explain using:

⚠️ Common Mistakes: Writing Linear Equations

Avoid these 3 frequent errors

🌍 Real-World Applications: Writing Linear Equations

See how this math is used in the real world

📝 Worked Example: Solving a Quadratic by Factoring

Problem:

Solve $x^2 - 5x + 6 = 0$ .

2Factor the quadratic

3Set each factor equal to zero

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📌 Related Topics in Linear Equations

Solving Linear Equations Graphing Linear Equations Linear Inequalities Absolute Value Equations Linear Equation Word Problems

❓ Frequently Asked Questions

What is Writing Linear Equations?▾

Slope-intercept form and point-slope form

How can I study Writing Linear Equations effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 3 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Writing Linear Equations study guide free?▾

Yes — all study notes, flashcards, and practice problems for Writing Linear Equations on Study Mondo are free to access. No account is needed.

What course covers Writing Linear Equations?▾

Writing Linear Equations is part of the Algebra 1 course on Study Mondo, specifically in the Linear Equations section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Writing Linear Equations?▾

Yes, this page includes 3 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Writing Linear Equations

Try the Interactive Version!

Writing Linear Equations (Slope-Intercept Form)

What is Slope-Intercept Form?

Why Use This Form?

Understanding Slope (m)

Understanding Y-Intercept (b)

Finding Slope from Two Points

Writing Equations Given Slope and Y-Intercept

Writing Equations Given Slope and One Point

Writing Equations Given Two Points

Writing Equations from a Graph

Parallel Lines

Perpendicular Lines

Converting to Slope-Intercept Form

Real-World Applications

Common Mistakes to Avoid

Problem-Solving Strategy

Quick Reference

Practice Tips

📚 Practice Problems

1Problem 1easy

❓ Question:

2Problem 2medium

❓ Question:

3Problem 3hard

❓ Question:

⚠️ Common Mistakes: Writing Linear Equations

🌍 Real-World Applications: Writing Linear Equations

📝 Worked Example: Solving a Quadratic by Factoring

Practice Lab

Practice with Flashcards

Browse All Topics

📌 Related Topics in Linear Equations

❓ Frequently Asked Questions