Problem 54 - Prealgebra

Question

CHALLENGE Where are all of the possible locations for the graph of \((x, y)\) if \(x=-2 ?\)

Step-by-Step Solution

Answer

The graph is a vertical line at \( x = -2 \) covering all possible \( y \)-values.

Step 1: Understanding the Constraint

The constraint given is that \( x = -2 \). This means we are focusing on a vertical line that passes through all points where the \( x \)-coordinate is -2, irrespective of the \( y \)-coordinate.

Step 2: Visualizing the Line

Since \( x = -2 \) represents a vertical line, this line extends infinitely in the positive and negative \( y \) directions. There are no restrictions on \( y \), meaning any value of \( y \) is possible. The line can be described by all points \((x, y)\) where \( x = -2 \).

Step 3: Determining Locations

The graph of \((x, y)\) where \( x = -2 \) can be located anywhere along the vertical line that passes through \( x = -2 \) on the coordinate plane. This includes points such as \((-2, 0)\), \((-2, 5)\), \((-2, -3)\), etc. There are infinitely many such points.

Step 4: Conclusion on Possible Locations

Since the vertical line \( x = -2 \) intersects the plane at every possible \( y \)-value, there are infinite possible locations for the graph of \((x, y)\). The line itself is the locus of all these points.

Key Concepts

Coordinate Plane

The coordinate plane is a flat, two-dimensional surface that contains two intersecting lines, known as axes. These axes are usually labeled as the x-axis, which is horizontal, and the y-axis, which is vertical. The point where they intersect is called the origin, and it is denoted as (0, 0). Each point on the plane is represented by a pair of numbers written as (x, y).

• The first number, x, indicates the horizontal position relative to the origin.
• The second number, y, indicates the vertical position relative to the origin.

Understanding the coordinate plane is crucial because most graphing begins with plotting points or lines on it. The plane allows us to visually represent relationships and solve problems in mathematics and the physical world.

Vertical Line

A vertical line is a straight line that moves up and down the coordinate plane. It has an interesting characteristic: every point on this line has the same x-coordinate. The general equation for a vertical line is x = a, where a is a constant number. When you graph a vertical line, you plot a straight path that cuts through every potential y-value, meaning:

• These lines are parallel to the y-axis.
• They never intercept the x-axis except when they are on the y-axis itself.

In the exercise, the vertical line is given by the equation x = -2. This indicates that all possible points lie along this line, no matter what the y-coordinate is.

Graphing Points

Graphing points on a coordinate plane involves marking a specific location based on its coordinates. For a point such as (-2, 3), you would: 1. Move 2 units left from the origin to get to the x-coordinate of -2. 2. Move 3 units up to reach the y-coordinate of 3. Each point represents a unique location, but when we deal with a vertical line like x = -2, several points such as (-2, 0), (-2, 5), and (-2, -3) can all be plotted on the same line. This showcases how multiple values of y share the same x-coordinate when dealing with vertical lines.

Infinite Solutions

Infinite solutions arise in mathematics when there are countless possible values that satisfy a given condition. In the case of the vertical line described by x = -2, since there is no restriction on y, - Every point on this line has x = -2 but y can be any real number. This indicates that the line is filled with infinite points, each representing a solution to the x = -2 equation.

• Infinite solutions are often seen in equations that represent lines covering entire ranges on the plane.
• In our particular exercise, this means the entire vertical line is a representation of infinite solutions.

Recognizing when infinite solutions occur is essential for understanding the breadth of an equation's possible outputs.