Lesson 4.1: Use the Rectangular Coordinate System

New Concept

The rectangular coordinate system makes algebra visual! Using ordered pairs $(x,y)$, we can plot points and represent relationships between two variables. You'll learn to plot points, identify coordinates, and check if a point is a solution to a linear equation.

What’s next

You've got the basics down. Next, you'll master plotting points and verifying solutions through a series of interactive examples and practice cards on our platform.

Property

An ordered pair, $(x, y)$, gives the coordinates of a point in a rectangular coordinate system.
The first number is the $x$-coordinate.
The second number is the $y$-coordinate.
The phrase 'ordered pair' means the order is important.
The point $(0, 0)$ is called the origin. It is the point where the $x$-axis and $y$-axis intersect.
The horizontal number line is called the $x$-axis.
The vertical number line is called the $y$-axis.
The $x$-axis and the $y$-axis together form the rectangular coordinate system.

Examples

• To plot the point $(2, 5)$, start at the origin, move 2 units to the right along the $x$-axis, and then 5 units up.

• To plot the point $(-4, 1)$, start at the origin, move 4 units to the left along the $x$-axis, and then 1 unit up.

Property

The axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

Points on the Axes
Points with a $y$-coordinate equal to 0 are on the $x$-axis, and have coordinates $(a, 0)$.
Points with an $x$-coordinate equal to 0 are on the $y$-axis, and have coordinates $(0, b)$.

Property

An equation of the form $Ax + By = C$, where $A$ and $B$ are not both zero, is called a linear equation in two variables.
A linear equation is in standard form when it is written $Ax + By = C$. Linear equations have infinitely many solutions.
An ordered pair $(x, y)$ is a solution of the linear equation $Ax + By = C$, if the equation is a true statement when the $x$- and $y$-values of the ordered pair are substituted into the equation.

Examples

• The equation $5x + 2y = 10$ is a linear equation in two variables, with $A=5$, $B=2$, and $C=10$.

• The equation $y = 4x - 3$ is a linear equation. It can be rewritten in standard form as $4x - y = 3$.

Property

An ordered pair $(x, y)$ is a solution of the linear equation $Ax + By = C$, if the equation is a true statement when the $x$- and $y$-values of the ordered pair are substituted into the equation.
To verify, substitute the $x$- and $y$-values from the ordered pair into the equation and determine if the result is a true statement.

Examples

• To check if $(2, 3)$ is a solution to $y = 2x - 1$, substitute $x=2$ and $y=3$: $3 = 2(2) - 1 \rightarrow 3 = 4 - 1 \rightarrow 3 = 3$. Yes, it is a solution.

• To check if $(5, 1)$ is a solution to $3x - 5y = 10$, substitute $x=5$ and $y=1$: $3(5) - 5(1) = 15 - 5 = 10$. Yes, it is a solution.

Property

To find a solution to a linear equation, you can pick any number you want to substitute into the equation for one variable, and then solve for the other variable. It is a good idea to choose a number that is easy to work with, such as 0.

When an equation is in standard form ($Ax + By = C$), it is often easy to find solutions by setting $x=0$ and then $y=0$.

Examples

• For the equation $y = 3x + 6$, let's pick $x=0$. Then $y = 3(0) + 6 = 6$. So, $(0, 6)$ is a solution.