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Chapter 7: Problem 85

Explain why \((5,-2)\) and \((-2,5)\) do not represent the same ordered pair.

Short Answer

Expert verified
The ordered pairs \((5, -2)\) and \((-2, 5)\) do not represent the same pair because in an ordered pair, the position of the elements matters. Switching the positions of the elements results in a different pair.

Step by step solution

01

Understanding an Ordered Pair

An ordered pair in mathematics is a pair of objects where the order of the objects is significant. It is written as \((a, b)\) where 'a' and 'b' are any numbers.
02

Compare the Pairs

Now let's look at our pairs, \((5, -2)\) and \((-2, 5)\). In each ordered pair, the position of 5 and -2 are different. In the ordered pair \((5, -2)\), '5' is in the first position and '-2' is in the second position, while in the ordered pair \((-2, 5)\), '-2' is in the first position and '5' is in the second position.
03

Conclusion

This proves that the ordered pairs \((5, -2)\) and \((-2, 5)\) are not the same, because in an ordered pair the position of the elements is significant. Changing the order of the elements makes them different pairs.

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Key Concepts

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

Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a crucial aspect of mathematics that deals with defining and representing geometric shapes in a numerical way using coordinates. Each point in the plane is determined by a pair of numbers, known as an ordered pair \(x, y\). These numbers indicate the location of a point on the Cartesian plane, which is formed by two perpendicular axes: the x-axis, running horizontally, and the y-axis, running vertically.
  • The first number in an ordered pair corresponds to the position along the x-axis, while the second number corresponds to the position along the y-axis.
  • This system enables precise pinpointing of locations of points and helps in understanding their relationships, such as distance and slope between points.
Understanding coordinate geometry is foundational for solving many mathematical problems and designing graphical models, as it provides a bridge between algebra and geometry.
Position Significance
In mathematics, especially in the context of ordered pairs, the position of elements holds significant importance. When we talk about ordered pairs \(a, b\), it's crucial to maintain the order: the first element (\(a\)) must always be the x-coordinate and the second element (\(b\)) must always be the y-coordinate. This sequence determines the precise location of the point on the coordinate plane.
  • Switching the order changes the point's location, representing entirely different points. For example, \( (5, -2) \) indicates a location entirely distinct from \((-2, 5)\) because the x and y coordinates have swapped places.
  • This illustration of position significance is essential in understanding why ordered pairs can't simply be rearranged without altering their meaning or effect.
Thus, preserving the order in ordered pairs is vital for accurate mathematical representation and analysis.
Mathematical Reasoning
Mathematical reasoning involves using logic and critical thinking to solve mathematical problems. When dealing with ordered pairs, reasoning helps us understand why the specific order is crucial.
  • Logical steps might include evaluating the function of each element within the pairing and its influence on plotting on the Cartesian plane.
  • Critical thinking aids in drawing conclusions, such as realizing that a change in the sequence of numbers results in a completely different point.
In our example, \( (5, -2) \) and \((-2, 5)\) look superficially similar but represent unique points due to the significance of order. By employing mathematical reasoning, you can conclude that reversing order isn't a mere swapping; it alters the entire spatial context of the data provided by the pair.

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