# Linear Equations

A **linear** equation is an equation for a straight **line**

### These are all linear equations:

y = 2x + 1 | ||

5x = 6 + 3y | ||

y/2 = 3 − x |

Let us look more closely at one example:

### Example: **y = 2x + 1** is a linear equation:

The graph of **y = 2x+1** is a straight line

- When x increases, y increases
**twice as fast**, so we need 2x - When x is 0, y is already 1. So +1 is also needed
- And so: y = 2x + 1

Here are some example values:

x | y = 2x + 1 |
---|---|

-1 |
y = 2 × (-1) + 1 = -1 |

0 |
y = 2 × 0 + 1 = 1 |

1 |
y = 2 × 1 + 1 = 3 |

2 |
y = 2 × 2 + 1 = 5 |

Check for yourself that those points are part of the line above!

## Different Forms

There are many ways of writing linear equations, but they usually have **constants** (like "2" or "c") and must have simple **variables** (like "x" or "y").

### Examples: These are linear equations:

y = 3x − 6 | ||

y − 2 = 3(x + 1) | ||

y + 2x − 2 = 0 | ||

5x = 6 | ||

y/2 = 3 |

**But** the variables (like "x" or "y") in Linear Equations do **NOT** have:

- Exponents (like the 2 in x
^{2}) - Square roots, cube roots, etc

### Examples: These are **NOT** linear equations:

y^{2} − 2 = 0 |
||

3√x − y = 6 | ||

x^{3}/2 = 16 |

## Slope-Intercept Form

The most common form is the slope-intercept equation of a straight line:

Slope (or Gradient) | Y Intercept |

### Example: y = 2x + 1

- Slope:
**m = 2** - Intercept:
**b = 1**

## Play With It !You can see the effect of different values of |

## Point-Slope Form

Another common one is the Point-Slope Form of the equation of a straight line:

y − y |

### Example: y − 3 = (¼)(x − 2)

It is in the form **y − y _{1} = m(x − x_{1})** where:

- y
_{1}= 3 - m = ¼
- x
_{1}= 2

## General Form

And there is also the General Form of the equation of a straight line:

Ax + By + C = 0 |

(A and B cannot both be 0) |

### Example: 3x + 2y − 4 = 0

It is in the form **Ax + By + C = 0** where:

- A = 3
- B = 2
- C = −4

There are other, less common forms as well.

## As a Function

Sometimes a linear equation is written as a function, with f(x) instead of y:

y = 2x − 3 |

f(x) = 2x − 3 |

These are the same! |

And functions are not always written using f(x):

y = 2x − 3 |

w(u) = 2u − 3 |

h(z) = 2z − 3 |

These are also the same! |

## The Identity Function

There is a special linear function called the "Identity Function":

f(x) = x

And here is its graph:

It makes a 45° (its slope is 1)

It is called "Identity" because what comes out is **identical** to what goes in:

In | Out |
---|---|

0 | 0 |

5 | 5 |

−2 | −2 |

...etc | ...etc |

## Constant Functions

Another special type of linear function is the Constant Function ... it is a horizontal line:

f(x) = C

No matter what value of "x", f(x) is always equal to some constant value.

## Using Linear Equations

You may like to read some of the things you can do with lines: