Lesson 1: Use the Rectangular Coordinate System

New Concept

The rectangular coordinate system is a grid formed by an x-axis and a y-axis. It allows us to plot points using ordered pairs $(x, y)$, identify locations on a graph, and find solutions to linear equations.

What’s next

You're now ready to master this system. Next, you will work through interactive examples and practice cards to plot points and identify their coordinates.

Property

The rectangular coordinate system is formed by two number lines, one horizontal and one vertical, that intersect at their zero points. The horizontal number line is the x-axis. The vertical number line is the y-axis. The axes divide a plane into four regions called quadrants. The point $(0, 0)$ is called the origin, where the axes intersect.

Examples

• The point $(4, 6)$ is in Quadrant I because both coordinates are positive. You move 4 units right and 6 units up from the origin.
• The point $(-2, -7)$ is in Quadrant III. You move 2 units left and 7 units down from the origin.
• The point $(-5, 3)$ is in Quadrant II. You move 5 units left and 3 units up from the origin.

Explanation

Think of the coordinate system as a map for numbers. The x-axis tells you how far to go left or right, and the y-axis tells you how far to go up or down. The four quadrants are the four zones created by the axes.

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.

Examples

• To plot $(3, 5)$, you start at the origin, move 3 units to the right along the x-axis, and then 5 units up.
• The point $(5, 3)$ is different. You move 5 units to the right and 3 units up. The order of the coordinates is crucial.
• The point $(-2, 1)$ means you move 2 units to the left (negative x direction) and 1 unit up.

Explanation

An ordered pair provides the exact 'address' of a point on the coordinate plane. The order matters: the first number is always for the horizontal x-axis, and the second is for the vertical y-axis. Changing the order changes the location.

Property

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 $(0, b)$. The point $(0, 0)$ is the origin.

Examples

• The point $(6, 0)$ has a y-coordinate of 0, so it lies directly on the x-axis.
• The point $(0, -4)$ has an x-coordinate of 0, so it is located on the y-axis.
• The point $(0, 0)$ is the origin, lying on both the x-axis and the y-axis.

Explanation

If one of the coordinates in an ordered pair is zero, the point isn't in a quadrant—it's on an axis! A zero for the y-value means no vertical movement, so the point lies on the x-axis. A zero for x means no horizontal movement.

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.

Examples

• The equation $4x - 2y = 10$ is a linear equation where $A=4$, $B=-2$, and $C=10$.
• $y = 6x - 2$ is also a linear equation. It can be rewritten in the standard form as $6x - y = 2$.
• $x = 9$ is a linear equation in two variables, which can be written as $1x + 0y = 9$.

Explanation

This is a special kind of equation involving two variables (like $x$ and $y$). It's called 'linear' because if you plot all its possible solutions on a graph, they form a perfectly straight line.

Property

An ordered pair $(x, y)$ is a solution to 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 find a solution, choose any value for one variable, substitute it into the equation, and solve for the other variable.

Examples

• To check if $(3, 1)$ is a solution to $2x + y = 7$, substitute the values: $2(3) + 1 = 6 + 1 = 7$. Since $7 = 7$, it is a solution.
• Find a solution to $y = 3x + 1$. Let's choose $x=2$. Substitute it in: $y = 3(2) + 1 = 7$. So, $(2, 7)$ is a solution.
• Find a solution to $4x - 2y = 10$. Let's choose $y=1$. Substitute it in: $4x - 2(1) = 10$, which means $4x = 12$, so $x=3$. The ordered pair $(3, 1)$ is a solution.

Explanation

A solution is an $(x, y)$ pair that makes the equation true. Think of it as a specific point that lies on the equation's line. Since the line is infinite, there are infinitely many solutions you can find for any linear equation.