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Chapter 3: Problem 74

Find at least five ordered pair solutions and graph them. $$ x=-54 $$

Short Answer

Expert verified
Ordered pairs: \((-54, -2), (-54, 0), (-54, 3), (-54, 7), (-54, 10)\).

Step by step solution

01

Understand the Equation

The given equation is \( x = -54 \). This means that for any value of \( y \) in the pair \((x, y)\), the value of \( x \) will always be \(-54\).
02

Choose Values for y

Since \( x \) is constant, you can choose any values for \( y \). Let's choose \( y = -2, 0, 3, 7, 10\).
03

Formulate Ordered Pairs

Combine each chosen \( y \) value with \( x = -54 \) to form ordered pairs: 1. \((-54, -2)\)2. \((-54, 0)\)3. \((-54, 3)\)4. \((-54, 7)\)5. \((-54, 10)\)
04

Graph Ordered Pairs

Plot the five ordered pairs on a Cartesian coordinate system. Since \( x = -54 \) is constant, all the points will align vertically along the line \( x = -54 \).

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Key Concepts

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

Ordered Pairs
An ordered pair is a fundamental concept in mathematics used to define a point in a two-dimensional space. It consists of two values written in a specific order, generally enclosed in parentheses like this:
  • The first value represents the x-coordinate, which is the horizontal position of the point.
  • The second value signifies the y-coordinate, dictating the vertical position.
Ordered pairs are crucial for plotting points on a grid because they give precise information about where a point is located.
For instance, in the pair \((-54, -2)\), \(-54\) is the x-coordinate and \(-2\) is the y-coordinate. Understanding ordered pairs helps in graphing and interpreting relationships between variables easily.
Cartesian Coordinate System
The Cartesian coordinate system is a grid-based framework used for graphing mathematical relationships. It consists of two perpendicular lines called axes:
  • The x-axis, which runs horizontally.
  • The y-axis, which runs vertically.
These axes intersect at a point known as the origin, marked by (0, 0). Each point in the system is represented by an ordered pair \((x, y)\), where the x-value determines the point's horizontal position and the y-value its vertical position.
This system allows for easy visualization of equations such as lines, curves, and other relationships in a two-dimensional space. It provides an intuitive method for understanding how different values interact and align visually on a graph.
Vertical Lines
Vertical lines are a special type of line in the Cartesian coordinate system characterized by having a constant x-value. This means that no matter what y-value is taken, the x-value remains the same.
For example, the equation \(x = -54\) describes a vertical line at \(-54\) on the x-axis.Such lines are plotted parallel to the y-axis and show a uniform, non-changing relationship horizontally across the grid. When plotting points like \((-54, -2), (-54, 0), (-54, 3)\), and so on, you'll notice all points lie directly on top of one another forming a vertical line.
Constant x-value
A constant x-value in an equation, such as \(x = -54\), has a distinct meaning in graphing. This type of equation indicates that regardless of what y-values you choose, the x-coordinate stays the same.It emphasizes that for every point on this line, the horizontal or x-position doesn't change, leading to a vertical line formation on the graph.Choosing different y-values simply moves the point up and down along the line, but the x-coordinate will always be \(-54\).This provides an excellent visual representation of a fixed position along the x-axis, showcasing how varying variables only affect the y-value.

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