Problem 32 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$x=-3$$

Step-by-Step Solution

Answer

The equation \( x = -3 \) forms a vertical line, with points like \((-3, -1)\), \((-3, 0)\), and \((-3, 1)\) lying on it.

Step 1: Understanding the Equation

The equation is given as \( x = -3 \). This is an equation of a vertical line in the coordinate plane, meaning for any point on this line, the x-coordinate is always \(-3\). The y-coordinate can be any real number.

Step 2: Selecting Y-Values

To find solutions, we choose different values for \( y \) while \( x \) remains \(-3\). Since the line runs vertically, typical choices might be \( y = -1, 0, \) and \( 1 \).

Step 3: Forming Coordinates

Using the chosen values of \( y \), the points can be formed: when \( y = -1 \), the point is \((-3, -1)\); when \( y = 0 \), the point is \((-3, 0)\); when \( y = 1 \), the point is \((-3, 1)\).

Step 4: Drawing the Graph

Plot the points \((-3, -1)\), \((-3, 0)\), and \((-3, 1)\) on the coordinate plane. Connect these points with a vertical line running parallel to the y-axis. The line represents all solutions to the equation \( x = -3 \).

Key Concepts

Understanding Vertical Lines

In the realm of graphing linear equations, a vertical line stands out due to its unique characteristics. A vertical line is represented by the equation \( x = a \), where \( a \) is a constant. This equation indicates that for every point along the line, the x-coordinate remains same as the constant \( a \), while the y-coordinate can vary.

This is different from lines with slopes, where both coordinates change together. Vertical lines have:

• No slope (the slope is undefined).
• A straight up-and-down orientation parallel to the y-axis.
• Equations solely dependent on a fixed x-value.

Understanding vertical lines is crucial as they define a specific set of solutions on a coordinate plane where all x-values are identical.

Navigating the Coordinate Plane

The coordinate plane is a two-dimensional space characterized by a horizontal axis (x-axis) and a vertical axis (y-axis), intersecting at a point called the origin \( (0,0) \).

Every point in this plane is defined by an ordered pair \( (x,y) \) that specifies its location relative to the axes.

When graphing equations, the coordinate plane serves as a visual guide to plot and connect points. Here are some key features:

• The x-axis runs left to right and determines the horizontal position of points.
• The y-axis runs up and down and determines the vertical position of points.
• Quadrants divide the plane into four regions, each hosting different combinations of positive and negative coordinates.

Understanding the coordinate plane helps students to logically organize and visualize solutions, particularly when drawing graphs to represent equations like \( x = -3 \).

Exploring Solutions of Equations

To fully understand the solutions to an equation, it is essential to know what it means for a pair of numbers to be a solution. A "solution" for an equation such as \( x = -3 \) refers to any point on the line where the equation holds true. In simpler terms, this means for this equation, any point \((x, y)\) that satisfies \( x = -3 \) is considered a solution.

Finding solutions involves selecting values strategically:

• For a vertical line equation, you choose various y-values while keeping x constant.
• Each pair formed \( (x, y) \) represents a legitimate point on the graph.

Understanding how to identify solutions and plot them plays a significant role in being able to correctly graph equations and is an essential skill in algebra.