A linear equation is every equation that can written in form

So these are examples of some linear equations:

*x + 2 = 0*, so*a = 1, b = 2*.*42x − 7 = 0*, so*a = 42, b = − 7*.*− 2x = 0*, so*a = − 2, b = 0*.

These are examples of equations that are not linear:

*x*because of the^{2}+ 3 = 0*x*.^{2}*7x + 2 = sin(x)*because of the*sin(x)*.

In other words, we can define the linear equation as an equation *f(x) = 0* where *f* is a linear function.

## Solving Simple Linear Equation

Let’s take as an example equation *2x − 16 = 0*. How should we solve it? Our goal is to find the value of *x* for which the equation holds. E. g. for *x = 10* it doesn’t hold since when we substitute *x* for *10* we get:

2 · 10 − 16 | = | 0 |

20 − 16 | = | 0 |

4 | ≠ | 0 |

It means the answer *x = 10* is not correct. But don’t cry, the actually is a single value of *x* for which it holds. We just have to find it. We can rearrange the equation by adding * + 16* to it:

2x − 16 | = | 0 |

2x | = | 16 |

Now we have the equation *2x = 16*. We can divide it by 2:

2x | = | 16 |

x | = | 8 |

And we have our result. We can verify it by substituting the *x* for *8*:

2 · 8 − 16 | = | 0 |

16 − 16 | = | 0 |

0 | = | 0 |

## The General Solution

We can see a pattern there. If we have a linear equation in the form *ax + b = 0*, we can always use the same approach. First, we subtract *b*:

ax + b | = | 0 |

ax | = | − b |

And then we divide by *a*:

Thus for each linear equation *ax + b = 0* we can find the *x* by letting *x = − b/a*.

## The Graphical Solution

From the chapter about linear function we know that a graph of each linear function is a straight line. If we find the *x* for which *ax + b = 0*, we get coordinates *(x, 0)*. These are coordinates of intersection point: a point where the graph of linear function intersects the horizontal axis. Let’s look back to our equation *2x − 16 = 0*. This is a graph of linear function *f(x) = 2x − 16*:

As you can see, the line intersects the horizontal axis in the point *(8, 0)*, which is exactly what have we have expected.