The Rectangular Coordinate System: Videos & Practice Problems

The rectangular coordinate system, or Cartesian plane, uses perpendicular x and y axes to plot ordered pairs $\mathrm{(x,\; y)}$, representing points in four quadrants. Solutions to two-variable linear equations, such as $x+\; 2y=\; 5$, are ordered pairs that satisfy the equation when substituted. Understanding how to graph points and verify solutions enhances comprehension of multivariable polynomials and their terms, including numerical coefficients and degrees. This foundational skill supports analyzing polynomial expressions and their behavior in standard form with descending powers.

Graphs & the Rectangular Coordinate System

In this lesson, we explore the rectangular coordinate system, also known as the Cartesian plane, which allows us to graph relationships between two variables. This system consists of two perpendicular number lines: the horizontal x-axis and the vertical y-axis, creating a two-dimensional grid. Points in this system are represented as ordered pairs, written in the form (x, y), where 'x' indicates the position along the x-axis and 'y' indicates the position along the y-axis.

To plot a point, you start at the origin, which is the point (0, 0) where the x-axis and y-axis intersect. For example, to plot the point (4, 3), you move 4 units to the right along the x-axis and then 3 units up along the y-axis. Conversely, for a point like (-3, 2), you would move 3 units to the left on the x-axis and then 2 units up on the y-axis. Understanding the signs of the coordinates is crucial: positive x-values are to the right of the origin, while negative x-values are to the left; similarly, positive y-values are above the origin, and negative y-values are below.

As you practice plotting points, you will encounter various combinations of positive and negative coordinates. For instance, the point (-2, -3) requires moving 2 units left and 3 units down, while (5, -4) involves moving 5 units right and then 4 units down. The origin (0, 0) serves as a reference point, and points like (0, -3) indicate movement straight down from the origin without any horizontal displacement.

The Cartesian plane is divided into four regions known as quadrants, which are numbered counterclockwise starting from the top right. Quadrant I contains positive x and y values, Quadrant II has negative x and positive y values, Quadrant III has negative x and y values, and Quadrant IV has positive x and negative y values. Understanding these quadrants helps in visualizing the location of points based on their coordinates.

By mastering the plotting of points and understanding the coordinate system, you will be well-equipped to analyze and represent relationships between two variables effectively.

Graphs & the Rectangular Coordinate System Example 1

In this lesson, we explore how to graph points on a coordinate plane and identify their respective quadrants. Each point is represented by an ordered pair, where the first number corresponds to the x-coordinate and the second number corresponds to the y-coordinate. To graph a point, you start at the origin (0,0) and move along the x-axis either to the right (for positive values) or to the left (for negative values), followed by moving vertically along the y-axis either up (for positive values) or down (for negative values).

For example, to graph the point (1, -2), you would move 1 unit to the right on the x-axis and then 2 units down on the y-axis, placing the point in the fourth quadrant. Similarly, for the point (5, 2), you would move 5 units to the right and 2 units up, locating it in the first quadrant. The point (-3, -4) requires moving 3 units to the left and 4 units down, which places it in the third quadrant. Lastly, the point (-4, 3) involves moving 4 units to the left and 3 units up, positioning it in the second quadrant.

Understanding the quadrants is essential: Quadrant I (top right) contains points where both coordinates are positive, Quadrant II (top left) has a negative x-coordinate and a positive y-coordinate, Quadrant III (bottom left) has both coordinates negative, and Quadrant IV (bottom right) has a positive x-coordinate and a negative y-coordinate. This systematic approach to graphing points and identifying their quadrants is fundamental in coordinate geometry.

In which quadrant is the following point located?
$(-\frac{25}{2},47)$

• A. $Q1$
• B. $Q2$
• C. $Q3$
• D. $Q4$

Answer: (B)

In which quadrant is the following point located?
$(248,-321)$

• A. $Q1$
• B. $Q2$
• C. $Q3$
• D. $Q4$

Answer: (D)

Graphs & the Rectangular Coordinate System Example 2

Plotting points on the Cartesian plane involves identifying the coordinates given in the form (x, y) and locating their exact positions by moving along the x-axis and y-axis. For example, to plot the point (3, 4), start at the origin (0, 0), move 3 units to the right along the x-axis because the x-coordinate is positive 3, and then move 4 units up along the y-axis since the y-coordinate is positive 4. The intersection of these movements marks the point (3, 4).

Similarly, for the point (-4, 2), begin at the origin, move 4 units to the left on the x-axis due to the negative x-coordinate, and then move 2 units up on the y-axis. The intersection here represents the point (-4, 2). For points with negative y-values, such as (-3, -5), move 3 units left on the x-axis and 5 units down on the y-axis. This method ensures accurate placement of points in all four quadrants of the Cartesian plane.

When plotting (5, -2), move 5 units right on the x-axis and 2 units down on the y-axis. This systematic approach of moving horizontally according to the x-coordinate and vertically according to the y-coordinate helps in visualizing and graphing ordered pairs effectively.

Understanding how to plot points is fundamental in coordinate geometry, as it forms the basis for graphing lines, curves, and shapes. The Cartesian coordinate system uses two perpendicular number lines—the x-axis (horizontal) and the y-axis (vertical)—to define the position of points in a plane. Each point is an ordered pair (x, y), where x represents the horizontal displacement and y the vertical displacement from the origin.

By practicing this method, students develop spatial reasoning and gain the ability to interpret and create graphs, which are essential skills in mathematics and related fields. This technique also lays the groundwork for understanding more complex concepts such as functions, transformations, and vector spaces.

Determining Whether an Ordered Pair is a Solution to an Equation

Understanding how to solve equations with two variables builds directly on the concept of solving one-variable equations. In a one-variable equation, such as x + 2 = 5, a value for x is considered a solution if substituting it into the equation results in a true statement. For example, plugging in x = 3 yields 3 + 2 = 5, which is true, confirming that x = 3 is a solution.

When extending this idea to two-variable equations, the solution is an ordered pair (x, y). This ordered pair satisfies the equation if, when both x and y are substituted into the equation, the resulting statement is true. For instance, consider the linear equation x + 2y = 5. If we test the ordered pair (3, 1), substituting x = 3 and y = 1 gives 3 + 2(1) = 3 + 2 = 5, which is true. Therefore, (3, 1) is a solution to the equation.

To verify whether an ordered pair is a solution, substitute the values of x and y into the equation and simplify. For example, testing (5, 0) in x + 2y = 5 results in 5 + 2(0) = 5 + 0 = 5, which is true, so this pair is a solution. However, testing (8, 2) yields 8 + 2(2) = 8 + 4 = 12, which does not equal 5, so this pair is not a solution. Another example is (-3, 4), which gives -3 + 2(4) = -3 + 8 = 5, confirming it as a solution.

This process highlights that solving two-variable equations involves checking ordered pairs by substitution to determine if they satisfy the equation. The key is recognizing that an ordered pair (x, y) is a solution if it makes the equation true when both values are plugged in. This method is a natural extension of solving one-variable equations and is fundamental in understanding linear equations and their graphs.

Given the equation $y-4=2(x+1)$, which of the following ordered pairs is a solution?

• A. $\left(-1,-4\right)$
• B. $\left(0,6\right)$
• C. $\left(1,4\right)$
• D. $\left(2,7\right)$

Answer: (B)

Given the equation $y=\frac{1}{2}x-\frac{3}{2}$, which of the following ordered pairs is a solution?

• A. $\left(4,\frac12\right)$
• B. $\left(2,\frac12\right)$
• C. $\left(6,0\right)$
• D. $\left(0,-3\right)$

Answer: (A)

Here’s what students ask on this topic:

What is the rectangular coordinate system and how do you plot points on it?

The rectangular coordinate system, also known as the Cartesian plane, consists of two perpendicular number lines: the horizontal x-axis and the vertical y-axis. These axes intersect at the origin, which is the point (0, 0). To plot a point, you use an ordered pair (x, y), where x represents the horizontal position and y represents the vertical position. Starting at the origin, move x units along the x-axis (right for positive, left for negative), then move y units along the y-axis (up for positive, down for negative). For example, to plot (4, 3), move 4 units right and 3 units up. This system divides the plane into four quadrants, which help identify the sign of the coordinates in each region.

How do you determine which quadrant a point lies in on the Cartesian plane?

The Cartesian plane is divided into four quadrants by the x-axis and y-axis. Quadrant I is the top-right region where both x and y are positive. Quadrant II is the top-left region where x is negative and y is positive. Quadrant III is the bottom-left region where both x and y are negative. Quadrant IV is the bottom-right region where x is positive and y is negative. To determine the quadrant of a point (x, y), check the signs of x and y: if both are positive, the point is in Quadrant I; if x is negative and y positive, Quadrant II; if both negative, Quadrant III; and if x positive and y negative, Quadrant IV.

How can you verify if an ordered pair is a solution to a two-variable linear equation?

To verify if an ordered pair (x, y) is a solution to a two-variable linear equation, substitute the x and y values into the equation and check if the resulting statement is true. For example, given the equation $x+2y=5$ and the ordered pair (3, 1), substitute x = 3 and y = 1: $3+2\times 1=5$. Simplify to get $3+2=5$, which is true. Therefore, (3, 1) is a solution. If the equation does not hold true after substitution, the ordered pair is not a solution.

What is the origin in the rectangular coordinate system and why is it important?

The origin is the point where the x-axis and y-axis intersect in the rectangular coordinate system, represented as (0, 0). It serves as the reference point for locating all other points on the plane. The origin divides the plane into four quadrants and separates positive and negative values on both axes. When plotting points, you start at the origin and move according to the x and y coordinates. Understanding the origin is essential because it helps in interpreting the position of points and understanding the structure of the coordinate system.

How do the x and y axes divide the Cartesian plane and what are their roles?

The x-axis is the horizontal number line, and the y-axis is the vertical number line in the Cartesian plane. They intersect at the origin (0, 0) and divide the plane into four quadrants. The x-axis represents the horizontal component of a point's position, while the y-axis represents the vertical component. Together, they allow us to describe any point in two dimensions using an ordered pair (x, y). Positive x values lie to the right of the origin, negative x values to the left; positive y values lie above the origin, negative y values below.