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Chapter 3: Problem 44

\(\mathrm{Do}(4,-1)\) and \((-1,4)\) represent the same ordered pair? Explain.

Short Answer

Expert verified
(4, -1) and (-1, 4) are not the same ordered pair because their x-coordinates and y-coordinates are different.

Step by step solution

01

- Understand Ordered Pairs

Ordered pairs consist of two elements: the first represents the x-coordinate and the second represents the y-coordinate. For two ordered pairs to be the same, both elements in the pairs must be identical.
02

- Compare the x-coordinates

Compare the x-coordinates of the given pairs. The first pair is \(4, -1\), with an x-coordinate of \(4\). The second pair is \((-1, 4)\), with an x-coordinate of \((-1)\). Since \4 \e -1\, the x-coordinates are different.
03

- Conclude the Comparison

Since the x-coordinates are different, the ordered pairs \(4, -1\) and \((-1, 4)\) are not the same. Both coordinates must match for ordered pairs to be equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-coordinate
In algebra, an ordered pair is a fundamental concept in coordinate geometry. The first number in this pair is called the x-coordinate.
This number tells you how far left or right to move along the x-axis.
  • If the x-coordinate is positive, you move to the right.
  • If it's negative, you move to the left.
For example, in the ordered pair \( (4, -1) \), the x-coordinate is 4, so you move 4 units to the right along the x-axis.
Understanding x-coordinates is essential for graphing points on a coordinate plane.
y-coordinate
The second number in an ordered pair is called the y-coordinate.
This number tells you how far up or down to move along the y-axis.
  • If the y-coordinate is positive, you move up.
  • If it's negative, you move down.
For example, in the ordered pair \( (4, -1) \), the y-coordinate is -1, so you move 1 unit down along the y-axis.
Like the x-coordinate, understanding y-coordinates is crucial for correctly plotting points on a graph.
comparison of ordered pairs
To determine if two ordered pairs are the same, both their x-coordinates and y-coordinates must match.
For instance, let's compare the pairs (4, -1) and (-1, 4). You start by comparing their x-coordinates:
  • First pair has x-coordinate 4.
  • Second pair has x-coordinate -1.
Since 4 is not equal to -1, the x-coordinates are different.
Next, you would look at the y-coordinates, but since the x-coordinates already did not match, the pairs are different.
Hence, (4, -1) and (-1, 4) are not the same ordered pairs because the x-coordinates alone are different.
algebra basics
Algebra often involves working with ordered pairs, especially in coordinate geometry and graphing.
An ordered pair \( (x, y) \) represents a specific point on a coordinate plane.
  • The x-coordinate indicates the horizontal position.
  • The y-coordinate points to the vertical position.
Ordered pairs are essential in forming relationships between variables and in graphing linear equations.
Understanding this basic concept lays the groundwork for more advanced topics in algebra.

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