Linear Equation

A linear equation is every equation that can written in form

ax + b = 0, where a, c∈ℝ and a≠0.

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:

  • x2 + 3 = 0 because of the x2.
  • 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
40

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:

\begin{eqnarray}ax&=&-b\\x&=&-\frac{b}{a}\end{eqnarray}

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:

Graph of the 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.