Graphical representation of linear equations

One of the things that we must be clear to understand the graphic representation of linear equations is knowing that it is a line. A line is a line that joins two points in space, in mathematics this is a coordinate system, which in our case is the Cartesian (two dimensions x, y), formed by the intersection of two real lines, called "x axis" and "y axis".

Cartesian axes

When we have a point on the Cartesian plane, it can be represented in (x, y):

Cartesian points

By joining two points we create a line, which is represented algebraically with the following equation:

y - y0 = m (x - x0) + b      (1)

Where, (x, y) and (x0, Y0) are points on the Cartesian plane, m is the slope or inclination of the line, with respect to the coordinate system. b is the point where the line intersects the y-axis.

Example: Graph the following linear equation 2x + 1 = 5

Observing the equation we can realize that 2x + 1 has the form of a line like (1), we could say that this is the line y = 2x + 1:

graphic representation of linear equations 1

It has slope 2 and cuts the y axis at 1.

Now, on the other side of equality in the equation we have the value 5, taking it to the form of the line (1), it would be y = 5:

graphic representation of linear equations 2

It has slope 0 and cuts the y axis at 5.

If we intercept these two lines, we will have the solution value to the linear equation:

graphic representation of linear equations 3

We observe that the point of intersection is (2, 5) and x = 2. If we solve the equation algebraically:

2x + 1 = 5

-1 + 2x + 1 = 5 - 1

2x = 4

2x / 2 = 4/2

x = 2

Consequently, we can solve linear equations in two ways, algebraically and graphically.