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