The Rectangular Coordinate System and Plotting Ordered Pairs

The Rectangular Coordinate System

Introduction to the Coordinate Plane

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane formed by the intersection of a horizontal axis (x-axis) and a vertical axis (y-axis). This system is fundamental in College Algebra for graphing equations and analyzing relationships between variables.

• Axes: The horizontal axis is called the x-axis, and the vertical axis is called the y-axis.

• Origin: The point where the axes intersect is called the origin, denoted as (0, 0).

• Quadrants: The plane is divided into four regions called quadrants, numbered I to IV in a counterclockwise direction starting from the upper right.

Ordered Pairs

An ordered pair (x, y) represents a point in the plane, where x is the horizontal position and y is the vertical position. Each ordered pair corresponds to a unique point on the coordinate plane.

• Notation: (x, y)

• Example: (2, 5) means move 2 units right and 5 units up from the origin.

• Quadrant Location:

  • Quadrant I: x > 0, y > 0

  • Quadrant II: x < 0, y > 0

  • Quadrant III: x < 0, y < 0

  • Quadrant IV: x > 0, y < 0

  • Points on axes: If x = 0 or y = 0, the point lies on the y-axis or x-axis, respectively.

Plotting Ordered Pairs

To plot an ordered pair (x, y):

1. Start at the origin (0, 0).

2. Move x units left or right (right if x > 0, left if x < 0).

3. From that position, move y units up or down (up if y > 0, down if y < 0).

4. Mark the point.

Example: Plotting the points (-2, 3), (0, 4), (2, 5), and (4, 0):

Point	Quadrant/Axis
(-2, 3)	Quadrant II
(0, 4)	y-axis
(2, 5)	Quadrant I
(4, 0)	x-axis

Equations in Two Variables

Definition and Solutions

An equation in two variables is a mathematical statement involving two unknowns, typically x and y. A solution to such an equation is any ordered pair (x, y) that makes the equation true when substituted.

• Example: For the equation x - 2y = 3, the pair (5, 1) is a solution because 5 - 2(1) = 3.

• To check if an ordered pair is a solution, substitute the values into the equation and verify if the statement is true.

Solving Linear Equations

Solving for a Variable

To solve a linear equation for a variable, isolate the variable using algebraic operations.

• Example: Solve $\frac{2}{x+4} = \frac{5}{x-2}$

Steps:

1. Cross-multiply: $2(x-2) = 5(x+4)$

2. Expand: $2x - 4 = 5x + 20$

3. Rearrange: $2x - 5x = 20 + 4$

4. Simplify: $-3x = 24$

5. Divide: $x = -8$

Graphing Linear Equations

Graph of a Linear Equation

The graph of a linear equation in two variables is a straight line. To graph a line, plot at least two points that satisfy the equation and draw a line through them.

• Example: The equation y = 2x + 1 can be graphed by plotting points such as (0, 1) and (1, 3).

Summary Table: Quadrants and Axes

Quadrant	x-value	y-value
I	+	+
II	-	+
III	-	-
IV	+	-
x-axis	any (except 0)	0
y-axis	0	any (except 0)

Key Terms

• Ordered Pair: A pair of numbers (x, y) representing a point in the plane.

• Quadrant: One of the four regions into which the coordinate plane is divided.

• Origin: The point (0, 0) where the axes intersect.

• Linear Equation: An equation whose graph is a straight line.

Practice Example

Given the equation x - 2y = 3, determine if the following ordered pairs are solutions:

• (5, 1): $5 - 2(1) = 3$ → $5 - 2 = 3$ → $3 = 3$ ✔️

• (2, -1): $2 - 2(-1) = 3$ → $2 + 2 = 4$ → $4 \neq 3$ ❌