We will now begin our study of the equation of the line, Cartesian systems, and the principles of analytic geometry.

The Principles of Analytic Geometry

When real numbers are introduced, it is usually said that these are points on a line

Based on this, Descartes had the genius idea of using two lines to represent points on a plane as a pair of coordinates (x, y)

The Equation of the Line

Using these concepts, it is now possible to consider a set of points on the plane to form curves, where the coordinate x corresponds to another coordinate y, and this correspondence rule is given by a function. This is where algebra penetrates geometry, giving birth to “Analytic Geometry.”

Geometrically, we understand a line as the curve that connects two points by traveling the shortest possible distance.

Geometrically, we understand a line as the curve that connects two points by traveling the shortest possible distance. Analyzing this, by virtue of Thales’ theorem, we will see that for every increment of the coordinate y, there is a corresponding increment of the coordinate x such that the quotient m=(y_2 - y_1)/(x_2 - x_1)=\Delta y / \Delta x will always be constant for any pair of points on the line. This is what we call the “slope of the line.”

Since the slope is the same for any pair of points on the line, if we consider points on the line with coordinates (x, y), (x_0, y_0), (x_1, y_1) and (x_2, y_2), we can write:

\displaystyle \frac{y - y_0}{x - x_0} = \frac{y_2 - y_1}{x_2 - x_1}

Which is the same as saying

\begin{matrix}y & = & \displaystyle \frac{y_2 - y_1}{x_2 - x_1} (x - x_0 ) + y_0 \\ \\ & = & \displaystyle \frac{\Delta y}{\Delta x} (x - x_0) + y_0 \end{matrix}

This is where the well-known equation of the line comes from

\color{red}{{y = m(x - x_0) + y_0}}

Here, the pair (x_0, y_0) is a fixed point, while the pair (x, y) is any point.

Example Exercises

1. Calculate the equation of the line that passes through the point (x_0, y_0) = (2, 3) with a slope of m = 3/2 [SOLUTION]
2. Calculate the equation of the line that passes through the point (x_0, y_0) = (1, 8) with a slope of m = 7/5 [SOLUTION]
3. Calculate the equation of the line that passes through the points (x_1, y_1) = (3, 5) and (x_2, y_2) = (1, -2) [SOLUTION]

How to Graph the Equation of the Line

We have already seen how to obtain the equation of the line from some graphical information; now we will follow the reverse path, obtaining the graphical representation from the equation of the line.

At the end of the day, the equation of the line always ends up being presented as follows:

y = mx + b

Where m = \Delta Y / \Delta x is the slope and b is the position coefficient. From this, we have the following figure

Example Exercise

1. Graph the line with the equation y = \displaystyle \frac{3}{4}x + 2 [SOLUTION]
2. Graph the line with the equation y = \displaystyle -\frac{2}{5}x + 6 [SOLUTION]

Application Problems for the Line Equation

The line can be used to solve problems that involve the direct relationship between two magnitudes, as in the following examples:

1. A vehicle with an initial position x_0 = 12[m] moves with a speed of v = 0.3[m/s]. What will its position be after 30[s]? [SOLUTION]
2. A person goes to the market and buys 1[kg] of apples, spending a total of 50 Z\$. That same day, the same person returned to the market to buy another 3[kg] of apples, spending a total of 60 Z\$.. What is the price of the apples, and what is the price of the tickets? [SOLUTION]

Line Intersections

Suppose we have two lines and we want to find the common point between them; that is, find the intersection of the lines. To solve these types of problems, we must solve a system of equations. To understand this better, let’s look at the following example.

Let’s consider the following lines:

L_1 \; : \; y= \displaystyle \frac{3}{2}x + 1

L_1 \; : \; y=\displaystyle -\frac{1}{3}x + 9

Where do these two lines intersect?

To solve this, we reason as follows:

Therefore, the intersection point between the lines is (x, y)= \displaystyle \left(\frac{48}{11}, \frac{83}{11} \right).

Example Application Problem for Line Intersections

For a party, a total of 600 tickets were sold with total revenue of \$1,300,000. Youth tickets were sold for \$1,000, and adult tickets for \$3,000. How many adults and youth attended the party? [SOLUTION]