The Tables of Ordered Pairs

Oct 13, 2025 by windailys.com 68 views

Introduction

In mathematics, the study of lines and their graphs is a fundamental concept in algebra and geometry. A line is a set of points that extend infinitely in two directions, and its graph is a visual representation of these points on a coordinate plane. In this article, we will explore the concept of ordered pairs and how they represent points on the graphs of two different lines.

What are Ordered Pairs?

An ordered pair is a pair of numbers that are written in a specific order, usually in the form (x, y). The first number, x, represents the x-coordinate of a point on a coordinate plane, and the second number, y, represents the y-coordinate. For example, the ordered pair (3, 4) represents a point that is 3 units to the right of the origin and 4 units above the origin.

The Tables of Ordered Pairs

The tables of ordered pairs represent some points on the graphs of two different lines. The first table represents the points on the graph of Line [a], and the second table represents the points on the graph of Line [b].

Line [a]

x	-11	-6	-1	4
y	81	51	21	-9

Line [b]

x	-11	-6	-1	4
y	27	17	7	1

Finding the Equation of a Line

To find the equation of a line, we need to determine the slope and the y-intercept. The slope of a line is a measure of how steep it is, and the y-intercept is the point where the line intersects the y-axis.

Calculating the Slope

The slope of a line can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.

Calculating the Y-Intercept

The y-intercept of a line is the point where the line intersects the y-axis. It can be calculated using the formula:

b = y - mx

where b is the y-intercept, and m is the slope.

Finding the Equation of Line [a]

To find the equation of Line [a], we need to calculate the slope and the y-intercept.

Calculating the Slope of Line [a]

Using the formula for the slope, we get:

m = (51 - 81) / (-6 - (-11)) = (-30) / 5 = -6

Calculating the Y-Intercept of Line [a]

Using the formula for the y-intercept, we get:

b = 81 - (-6)(-11) = 81 - 66 = 15

Finding the Equation of Line [b]

To find the equation of Line [b], we need to calculate the slope and the y-intercept.

Calculating the Slope of Line [b]

Using the formula for the slope, we get:

m = (17 - 27) / (-6 - (-11)) = (-10) / 5 = -2

Calculating the Y-Intercept of Line [b]

Using the formula for the y-intercept, we get:

b = 27 - (-2)(-11) = 27 - 22 = 5

The Equations of the Lines

The equations of the lines are:

Line [a]: y = -6x + 15 Line [b]: y = -2x + 5

Conclusion

In conclusion, the tables of ordered pairs represent some points on the graphs of two different lines. By calculating the slope and the y-intercept, we can find the equation of each line. The equations of the lines are y = -6x + 15 and y = -2x + 5.

References

[1] Algebra and Geometry, by Michael Artin
[2] Calculus, by Michael Spivak

Introduction

In our previous article, we explored the concept of ordered pairs and how they represent points on the graphs of two different lines. We also calculated the equations of the lines using the slope and y-intercept. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the difference between a line and a graph?

A: A line is a set of points that extend infinitely in two directions, while a graph is a visual representation of these points on a coordinate plane.

Q: How do I determine the slope of a line?

A: To determine the slope of a line, you can use the formula: m = (y2 - y1) / (x2 - x1), where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.

Q: How do I calculate the y-intercept of a line?

A: To calculate the y-intercept of a line, you can use the formula: b = y - mx, where b is the y-intercept, and m is the slope.

Q: What is the equation of a line?

A: The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of the points on the line. It is usually written in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I find the equation of a line given two points?

A: To find the equation of a line given two points, you can use the slope formula to calculate the slope, and then use the y-intercept formula to calculate the y-intercept.

Q: What is the difference between a linear equation and a non-linear equation?

A: A linear equation is an equation that can be written in the form y = mx + b, where m is the slope and b is the y-intercept. A non-linear equation is an equation that cannot be written in this form.

Q: How do I graph a line?

A: To graph a line, you can use a coordinate plane and plot the points on the line. You can also use a graphing calculator or a computer program to graph the line.

Q: What is the significance of the y-intercept in a linear equation?

A: The y-intercept in a linear equation represents the point where the line intersects the y-axis. It is an important concept in linear algebra and is used to determine the equation of a line.

Q: How do I determine the equation of a line given the y-intercept and the slope?

A: To determine the equation of a line given the y-intercept and the slope, you can use the equation y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the difference between a horizontal line and a vertical line?

A: A horizontal line is a line that is parallel to the x-axis, while a vertical line is a line that is parallel to the y-axis.

Q: How do I graph a horizontal line?

A: To graph a horizontal line, you can plot a point on the line and draw a horizontal line through it.

Q: How do I graph a vertical line?

A: To graph a vertical line, you can plot a point on the line and draw a vertical line through it.

Conclusion

In conclusion, the tables of ordered pairs represent some points on the graphs of two different lines. By calculating the slope and the y-intercept, we can find the equation of each line. We also answered some frequently asked questions related to the topic.

References

[1] Algebra and Geometry, by Michael Artin
[2] Calculus, by Michael Spivak