Problem 19 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ y=0 $$

Step-by-Step Solution

Answer

The graph of \(y=0\) is a horizontal line along the \(x\)-axis.

Step 1: Understand the Equation

The equation we have is \(y=0\), which indicates that the value of \(y\) is constant and equal to zero, regardless of the value of \(x\). This represents a specific kind of line in the coordinate plane: a horizontal line.

Step 2: Identify Key Characteristics

A horizontal line at \(y=0\) means the line passes through all points where the \(y\)-coordinate is zero. In simpler terms, the line lies along the \(x\)-axis, which is the horizontal axis of the graph.

Step 3: Graph the Line

Since \(y=0\) is a horizontal line along the \(x\)-axis, plot the line by drawing a straight line that runs parallel to the \(x\)-axis. The line will pass through points like (0,0), (1,0), (-1,0), and any other point where \(y=0\).

Step 4: Finalize the Graph

Make sure your graph is extended appropriately along the \(x\)-axis to show that the line continues indefinitely in both the positive and negative directions, always remaining on the \(x\)-axis.

Key Concepts

Graphing Linear Equations

Linear equations are equations where the highest power of the variable is 1, often taking the form \(y = mx + b\). Here, \(m\) is the slope, and \(b\) is the y-intercept of the line. Graphing these equations involves plotting points on a graph that satisfy the equation, then drawing the line through these points.
To effectively graph any linear equation:

• Identify the y-intercept \((b)\). This is where the line crosses the y-axis. Plot this point first.
• Determine the slope \((m)\), which indicates how steep the line is. The slope is a ratio that describes the change in \(y\) for a unit change in \(x\). If \(m = 2\), rise 2 units up for every 1 unit you move right.
• With the y-intercept plotted and the slope calculated, draw a line through this point using the slope to guide the direction.

Linear equations can describe vertical, horizontal, or slanted lines depending on the slope and intercept.

Horizontal Lines

A horizontal line is a special type of linear equation. It appears as \(y = c\), where \(c\) is a constant. This indicates that no matter the value of \(x\), the value of \(y\) is always \(c\).
When graphing horizontal lines:

• The line runs parallel to the x-axis. Thus, it means there is no vertical change; the slope \(m\) is 0.
• For example, in the equation \(y = 0\), the line stretches along the x-axis. It includes all points like \((1, 0)\), \((-1, 0)\), etc.
• Horizontal lines are easy to recognize since they are always parallel to the base of the graph.

Horizontal lines depict scenarios where the dependent variable \(y\) remains constant irrespective of changes in the independent variable \(x\).

Coordinate Plane

The coordinate plane is a two-dimensional surface formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants and intersect at the origin \((0, 0)\).
To understand its properties:

• The x-axis is the horizontal line where \(y = 0\).
• The y-axis is the vertical line where \(x = 0\).
• Each point on the plane is represented by a pair of coordinates \((x, y)\), indicating distances from the axes.
• For instance, \((3, -2)\) is a point located 3 units to the right of the origin and 2 units down from the x-axis.

By plotting points and lines on the coordinate plane, you can visualize simple relationships in algebra, such as those formed by linear equations.