Forms of Linear Equations

Understanding How to Represent Lines in Different Ways

If you’ve read the earlier articles on Algebraic Expressions and Equations, Graphing Linear Equations, and Slope and Rate of Change, you already know how powerful linear equations are in algebra. These equations can describe trends, make predictions, and model real-world situations. But did you know there’s more than one way to write a linear equation?

In this article, we’ll explore the three most common forms of linear equations:

1. Slope–Intercept Form
2. Point–Slope Form
3. Standard Form

Each form has a different purpose, and knowing how to switch between them will strengthen your understanding of linear relationships.

1. Slope–Intercept Form

This is the most familiar form of a linear equation:

y = mx + b

• m is the slope (rate of change)
• b is the y-intercept (where the line crosses the y-axis)

Example:

Suppose a babysitter earns $10 per hour, plus $5 for travel. The total earnings E after h hours can be modeled as:

E = 10h + 5

This is in slope–intercept form, where the slope is 10 (dollars per hour) and the y-intercept is 5 (a fixed starting amount).

When to Use It:

• When you know the slope and y-intercept
• When graphing is your goal
• When analyzing how one variable affects another

2. Point–Slope Form

Sometimes, instead of the y-intercept, you’re given a point on the line and the slope. That’s when you use point–slope form:

y - y₁ = m(x - x₁)

• (x₁, y₁) is a point on the line
• m is the slope

Example:

A line passes through the point (3, 2) and has a slope of 4:

y - 2 = 4(x - 3)

You can simplify it to slope–intercept form:

y = 4x - 10

When to Use It:

• When you know the slope and a point on the line
• When you want to write an equation quickly without needing the y-intercept

3. Standard Form

The standard form of a linear equation looks like this:

Ax + By = C

• A, B, and C are integers
• A should be non-negative
• A and B should not both be zero

Example:

The equation:

3x + 4y = 12

is in standard form. You can rearrange it into slope–intercept form by solving for y:

4y = -3x + 12
y = -3/4x + 3

When to Use It:

• When working with systems of equations
• When dealing with integer values
• When graphing by intercepts

Comparing the Forms

Form	Structure	Best When You Know…
Slope–Intercept	y = mx + b	Slope and y-intercept
Point–Slope	y – y₁ = m(x – x₁)	Slope and any point
Standard	Ax + By = C	Integer values or intercepts

Practice Examples

1. Convert 5x + 2y = 10 to slope–intercept form.
   
   Answer: y = -5/2x + 5
   
2. Write the equation of a line through (–1, 4) with a slope of –3 in point–slope form.
   
   Answer: y - 4 = -3(x + 1)
   
3. Graph y = 2x - 1. Start at (0, –1) and use slope 2 (rise 2, run 1).
4. Convert y = 1/2x - 6 to standard form.
   
   Answer: x - 2y = 12
   
5. 
   Write a standard form equation for a line with slope 3 and y-intercept 2.
   
   Answer: Start with y = 3x + 2 → -3x + y = 2 → 3x - y = -2
   

Why This Matters

Understanding the forms of linear equations helps you move between problems easily. Sometimes you’ll be given a graph, other times a story problem, or just two points. Choosing the right form helps you write and solve equations efficiently.

As we continue into more advanced algebra topics like systems of equations or real-world modeling, you’ll find these forms give you tools to analyze, plan, and make predictions.

Next up? We’ll explore how to use two equations together — in systems of linear equations — and what happens when lines intersect, overlap, or never meet.