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, supporting skills in analyzing polynomial degrees, coefficients, and standard form expressions.

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 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.

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, and then move 4 units up along the y-axis. This intersection marks the location of the point.

Similarly, for the point (-4, 2), begin at the origin, move 4 units to the left (since the x-coordinate is negative), and then move 2 units up. The intersection of these movements is where the point (-4, 2) is plotted.

When plotting points with negative y-values, such as (-3, -5), start by moving 3 units left on the x-axis, then move 5 units down on the y-axis. The intersection of these positions marks the point.

For points like (5, -2), move 5 units right on the x-axis and 2 units down on the y-axis to find the correct location.

This method of plotting ordered pairs (x, y) relies on understanding the Cartesian coordinate system, where the x-axis represents horizontal movement and the y-axis represents vertical movement. Drawing imaginary lines or using grid lines to see where the x and y values intersect can help accurately place each point.

Mastering this skill is fundamental for graphing functions, analyzing geometric shapes, and solving problems involving spatial relationships in mathematics.

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, when x = 3, substituting gives 3 + 2 = 5, which is true, confirming that x = 3 is a solution.

Extending this idea to two-variable equations involves working with ordered pairs in the form (x, y). An ordered pair is a solution to a two-variable equation if, when the values of 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 yields:

\[3 + 2 \times 1 = 3 + 2 = 5\]

Since the equation holds true, (3, 1) is a solution to the equation.

To verify whether other ordered pairs satisfy the equation, substitute their values similarly. For example, testing (5, 0):

\[5 + 2 \times 0 = 5 + 0 = 5\]

This is true, so (5, 0) is a solution. However, testing (8, 2):

\[8 + 2 \times 2 = 8 + 4 = 12\]

Since 12 does not equal 5, (8, 2) is not a solution. Another example is (-3, 4):

\[-3 + 2 \times 4 = -3 + 8 = 5\]

This is true, so (-3, 4) is a solution.

This process highlights that solving two-variable equations involves verifying whether ordered pairs satisfy the equation by substitution, similar to how single-variable equations are solved. Recognizing that an ordered pair (x, y) is a solution if it makes the equation true is fundamental in algebra, especially when working with linear equations. This understanding lays the groundwork for graphing equations and exploring systems of equations, where solutions correspond to points on a coordinate plane.

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). The point where you end up is the location of the ordered pair on the plane. This system divides the plane into four quadrants, numbered counterclockwise starting from the top right.

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

Quadrants are the four regions created by the intersection of the x-axis and y-axis on the Cartesian plane. The quadrants are numbered counterclockwise starting from the top right. Quadrant 1 contains points where both x and y are positive. Quadrant 2 contains points where x is negative and y is positive. Quadrant 3 contains points where both x and y are negative. Quadrant 4 contains points where x is positive and y is negative. To determine the quadrant of a point, look at the signs of its coordinates (x, y) and match them to the corresponding quadrant.

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 significance of the origin in the rectangular coordinate system?

The origin is the point where the x-axis and y-axis intersect, represented by the ordered pair (0, 0). It serves as the reference point for locating all other points on the Cartesian plane. The origin divides the plane into four quadrants and separates positive and negative values on both axes. When plotting points, you start from the origin and move according to the x and y coordinates. Understanding the origin is essential for graphing points, interpreting coordinates, and analyzing the relationships between variables in two-dimensional space.

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; together, they form the Cartesian plane. These two axes 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. Each point on the plane is described by an ordered pair (x, y), where x is the distance along the x-axis and y is the distance along the y-axis. The axes help in graphing points, understanding spatial relationships, and analyzing equations involving two variables.