In these lessons, we will learn the various forms of linear equations and how to convert between the different forms.

**Related Pages**

Linear Equations In Two Variables

Graphing Linear Equations

Coordinate Geometry

Linear Equations or Equations of Straight Lines can be written in different forms.

We shall look at:

- the slope-intercept form for the equation of a line.
- the point-slope form for the equation of a line.
- the general form for the equation of a line.
- the standard form for the equation of a line.
- how to convert between the different forms of linear equations.

The following table gives the Forms of Linear Equations. Scroll down the page for examples
and solutions.

**y = mx + b**

where m is the slope of the line and b is the y-intercept.

The y-intercept is the y-coordinate of the location where line crosses the y axis. This is the point when x = 0 and y = b.

Vertical lines, having undefined slope, cannot be represented by this form.

The slope-intercept form is useful when we are given the slope and y-intercept of a line and we need to write an equation for the line.

It is also useful because we can read the slope and y-intercept from the equation. Often, when we are given equations in other forms, we can rewrite it in slope-intercept form to get the slope and y-intercept.

The slope-intercept form is also useful when we need to draw the line on a graph.

More examples, videos and explanations about the slope-intercept form and how it can be used.

**y − y _{1} = m(x − x_{1})**

where m is the slope of the line and (x

The point-slope form shows that the difference in the y-coordinate between two points on a line is proportional to the difference in the x coordinate. The proportionality constant is the slope of the line, m.

The point-slope form is useful when we are given a point on a line and the slope and we need to get the equation of the line.

**How to find the equation of a line using point-slope form when given two points?**

**Example:**

Find the equation of the line that goes through (-3,5),(2,8)

**Ax + By + C = 0**

where A or B can be zero, but not both at the same time.

The equation is usually written so that A ≥ 0. The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form.

If B is not zero, then the slope of the line is −A/B and the y-intercept is −C/B.

If B is zero, then the line Ax + C = 0 is a vertical line with a x-intercept of −C/A.

If A is zero, then the line By + C = 0 is a horizontal line with a y-intercept of −C/B.

The General Form is useful when we want to write equations for vertical lines which is not possible in slope-intercept form or point-slope form. For example, 2x + 5 = 0.

**Ax + By = C**

where A or B can be zero, but not both at the same time. A, B and C are integers and A ≥ 0. A,
B and C have no common factors other than 1.

When is Standard Form useful?

- When we want to solve systems of linear equations.
- When we want to write equations for vertical lines which is not possible in slope-intercept form or point-slope form. For example, 2x = 5.
- Simplifies finding parallel and perpendicular lines.

**How to find parallel lines using Standard Form?**

**Example:**

Find the equation of the line parallel to 8x - 9y = 3 and passing through (2,3)

**How to find perpendicular lines using Standard Form?**

**Example:**

- Find the equation of the line perpendicular to 8x - 9y = 3 and passing through (-5,7).
- Find the equation of the line parallel to y = 5 and passing through (-71,89).
- Find the equation of the line perpendicular to y = 5 and passing through (-6,-4).

**How to determine the equation of a line in standard form given two points?**

This video lesson shows how to determine the equation using slope-intercept form and then how to write the equation in standard form.

**Example:**

Determine the equation of the line passing through (-1,1) and (1,7). Write the line in standard form.

We will look at how to convert between the different forms of equation.

**How to convert from Standard Form to Slope Intercept Form?**

**Example:**

Convert the following to slope-intercept form 2x - 3y + 6 = 0

**How to convert the point-slope form of the equation of a line to slope-intercept form?**

**How to convert between point-slope form, slope-intercept form and standard form?**

**Example:**

A line passes through the points (-3,6) and (6,0). Find the equation of this line in:

a) Point-slope form

b) Slope-intercept form

c) Standard form

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