# Equation of a Straight Line — Forms & Examples

TL;DR  
The equation of a straight line is a relationship between xxx and yyy that every point on the line satisfies, written most commonly as y = mx + c (slope-intercept) or Ax + By + C = 0 (standard form). This article covers all five major forms — slope-intercept, point-slope, standard, intercept, and two-point — when to use each, how to convert between them, and six worked examples.

## What Is The Equation Of A Straight Line?  
The **equation of a straight line** is an algebraic relationship between the coordinates xxx and yyy such that a point (x,y) lies on the line if and only if its coordinates satisfy the equation. Every straight line has such an equation, and it is always **first-degree** — the variables xxx and yyy appear only to the power 1, never squared or higher.

That first-degree feature is what makes the graph straight. The moment a term like x^2 appears, the graph curves. So "linear equation" and "equation of a straight line" describe the same thing: a relationship whose graph is a [straight line](/content/math/geometry/straight-line/index.html). What changes from one form to the next is not the line — it is which feature of the line the equation puts front and center.

## What Are The Forms Of The Equation Of A Straight Line?  
There are five forms you will meet most often. Each is best when you know a particular piece of information about the line.

### Slope-intercept form  
$$y=mx+c$$  
Here **m** is the slope and **c** is the y-intercept (the y-value where the line crosses the y-axis). This is the most-used form because it reads the slope and intercept straight off. Use it when you know the slope and where the line meets the y-axis. It is covered in depth in [slope-intercept form](/content/math/geometry/slope-intercept-form-of-a-line/index.html) and the related [y = mx + b](/content/math/geometry/y-mx-b/index.html) page.

### Point-slope form  
$$y−y_1=m(x−x_1)$$  
Here **m** is the slope and **(x1,y1)** is a known point on the line. Use it when you know a point and the slope but not the intercept — see [point slope form](/content/math/geometry/point-slope-form/index.html) for the full treatment.

### Two-point form  
$$y−y_1=rac{y_2−y_1}{x_2−x_1}(x−x_1)$$  
This is point-slope form with the slope already written out from two points (x1,y1) and (x2,y2). Use it when all you are given is two points. It comes directly from [finding the slope from two points](/content/math/geometry/finding-slope-from-two-points/index.html).

### Standard (general) form  
$$Ax+By+C=0$$  
Here A, B, and C are constants, with A and B not both zero. Use it as a tidy, sign-balanced form that handles every line — including vertical ones, which slope-intercept form cannot.

### Intercept form  
$$\frac{x}{a} + \frac{y}{b} = 1$$  
Here **a** is the x-intercept and **b** is the y-intercept. Use it when you know where the line crosses both axes — see [intercept form](/content/math/geometry/intercept-form/index.html) for the derivation.

## How Do You Choose Which Form To Use?  
The honest answer is: match the form to what you are given. Know the slope and the y-intercept? Slope-intercept. Know a point and a slope? Point-slope. Know two points? Two-point (or compute the slope and use point-slope). Know both intercepts? Intercept form. Need to handle a vertical line or want integer coefficients? Standard form.

I will be honest — when I first learned these, I assumed I had to pick "the right one" and that choosing wrong would give a wrong answer. It will not. Every form describes the same line, and any one converts into any other with algebra. The choice is about convenience, not correctness.

## Examples of the Equation of a Straight Line  
### Example 1  
**Write the equation of the line with slope 2 and y-intercept −3.**  
Use slope-intercept form $$y=mx+c$$ with m=2 and c=−3:
$$y=2x−3$$  
That is the equation.

### Example 2  
**Find the equation of the line through (1,4) and (3,10).**  
Wrong path first. A student jumps to slope-intercept form and guesses c=4 because the first point's y-value is 4, writing $$y=mx+4$$. But 4 is the y-value at x=1, not at x=0. Substituting the point (1,4) would only give c after the slope is known.

Correct. Find the slope first:
$$m=\frac{10−4}{3−1}=\frac{6}{2}=3$$  
Use point-slope form with (1,4):
$$y−4=3(x−1)$$  
Distribute and simplify:
$$y−4=3x−3$$  
$$y=3x+1$$  
So the y-intercept is 1, not 4.

### Example 3  
**Convert 2x+3y−6=0 to slope-intercept form and state its slope.**  
Isolate the y term:
$$3y=−2x+6$$  
Divide every term by 3:
$$y=−\frac{2}{3}x+2$$  
The slope is −\frac{2}{3} and the y-intercept is 2.

### Example 4  
**A line has x-intercept 4 and y-intercept 2. Write it in intercept form, then in standard form.**  
Use intercept form with a=4, b=2:
$$\frac{x}{4} + \frac{y}{2} = 1$$  
Multiply through by 4 to clear denominators:
$$x+2y=4$$  
Move everything to one side for standard form:
$$x + 2y−4=0$$

### Example 5  
**The cost of a notebook plus a pen is modelled by a line. Two pens and a notebook cost 8; four pens and a notebook cost 12. If a pen costs x and a notebook costs y, write the equation relating one purchase pattern.**  
Take the two purchase conditions as points where x = pen count and the line tracks cost. Using the cost points (2,8) and (4,12):
$$m=\frac{12−8}{4−2}=\frac{4}{2}=2$$  
So each extra pen adds $2. Using point-slope form with (2, 8):
$$y−8=2(x−2)$$  
$$y=2x+4$$  
The constant $4 is the notebook's cost (the value when $x = 0$ pens).

### Example 6  
**Write the equation of the vertical line through (5,−2) and explain why slope-intercept form cannot express it.**  
A [vertical line](/content/math/geometry/vertical-line/index.html) has the equation $$x=a$$, where a is the shared x-coordinate:
$$x=5$$  
Slope-intercept form $$y=mx+c$$ requires a numerical slope, but a vertical line's slope is [undefined](/content/math/geometry/undefined-slope/index.html). Standard form handles it, though: $$x=5$$ fits $$Ax + By + C = 0$$ with B=0. This is exactly why standard form exists — it covers the lines the slope-based forms cannot.

## Why The Equation Of A Straight Line Matters  
Reducing a line to an equation is one of the foundational moves of mathematics, and it pays off far beyond geometry class.  
- **Coordinate geometry's central tool.**  
- **Linear models everywhere.**  
- **The base case for everything curved.**

## Where Students Trip Up On The Equation Of A Straight Line  
### Mistake 1: Reading a point's y-coordinate as the y-intercept  
**Where it slips in:** When given a point that is not on the y-axis.  
**The correct way:** The y-intercept c is the y-value only when x=0. For any other point, find the slope first, then solve for c by substituting the point.

### Mistake 2: Forcing every line into slope-intercept form  
**Where it slips in:** On vertical lines.  
**The correct way:** A vertical line is $$x=a$$ — no slope-intercept form exists for it because its slope is undefined.

### Mistake 3: Sign and coefficient errors when converting to standard form  
**Where it slips in:** When moving terms across the equals sign.  
**The correct way:** Move terms one at a time, tracking each sign, and clear denominators by multiplying every term.

## Key Takeaways  
- The **equation of a straight line** is a first-degree relationship between xxx and yyy that every point on the line satisfies.  
- The five common forms are slope-intercept, point-slope, two-point, standard, and intercept — all describing the same line.  
- Choose the form that matches what you know: slope and intercept, a point and slope, two points, or both intercepts.  
- Slope-intercept form cannot express a vertical line; standard form can.  
- Every form converts to every other with algebra, so the choice is about convenience, not correctness.