# Equations Of Lines

## Linear Equations as y = mx + c

The most commonly used form of a linear equation is *y = mx + c*

where

*m* is the gradient

*c* is the y-intercept.

## Example One - y = mx + c

Find the gradient *m* and the y-intercept *c* of the following equations:

(a) y = 3x + 4

(b) y = 5x – 2

(c) y = x

**Answer:**

(a) gradient = 3, y-intercept = 4

(b) gradient = 5, y-intercept = –2

(c) gradient = 1, y-intercept = 0

## Linear Equations in Standard Form

The standard form of linear equation is *ax + by = c*

In standard form, note that *c* in this form is **not** the y-intercept.

## Example Two - Standard Form

Write the equation y = 3x + 5 in standard form.

**Answer:**

y = 3x + 5

–3x + y = 5

*Traditionally in standard form, the coefficient of the x is always a positive number. To achieve this, multiply the whole equation by –1.*

3x – y = –5

## Example Three - Find Linear Equations

Use the formula: *y – y*_{1} = m (x – x_{1})

The gradient of a line is 2. The line passes through the point (4,5). What is the equation of this line?

**Answer:**

(x_{1}, y_{1}) = (4,5)

m = 2

y – y_{1} = m ( x – x_{1} )

y – 5 = 2 ( x – 4 )

y – 5 = 2x – 8

y = 2x – 8 + 5

y = 2x – 3

## Example Four - Find Linear Equations

A line passes through the points (4,5) and (7, 16). What is the equation of this line?

*(Hint: Find the gradient first.)*

**Answer:**

(x_{1}, y_{1}) = (4,5)

(x_{2}, y_{2}) = (7,16)

m = | y_{2} – y_{1} | = | 16 – 5 | = | 9 | = 3 |

x_{2} – x_{1} | 7 – 4 | 3 |

y – y_{1} = m ( x – x_{1} )

y – 5 = 3 ( x – 4 )

y – 5 = 3x – 12

y = 3x – 12 + 5

y = 3x – 7