Problem 25 - Algebra 1: Concepts and Skills

Question

Graph the equation. $$ y=8 $$

Step-by-Step Solution

Answer

The equation \(y = 8\) is a horizontal line crossing the y-axis at \(y = 8\).

Step 1: Identify the type of line

In the given equation, \(y = 8\), there are no x-terms. This means the line represented is a horizontal line. A horizontal line has the equation of form \(y = c\), where \(c\) is a constant and represents the y-coordinate of each point on the line.

Step 2: Plot the line

On a graph, plot a horizontal line crossing the y-axis at the point where \(y = 8\). This line will stay at the level of the 8th unit of the y-axis, running from negative infinity to positive infinity along the x-axis.

Key Concepts

Horizontal Lines

A horizontal line is a straight line that runs left to right and remains parallel to the x-axis. It is characterized by having a constant y-value across all of its points. In the equation format, a horizontal line is expressed as \(y = c\), where \(c\) denotes a constant. This means that no matter what value \(x\) takes, \(y\) remains the same. This is why in the equation \(y = 8\), the line is horizontal. This line will be a flat, unchanging path that simply crosses through multiple \(x\) values at the same \(y\) value.

Some key aspects of horizontal lines include:

• They have a slope of zero, which indicates no steepness or incline.
• In graphing, they are simple to identify due to their parallel position to the x-axis.
• Horizontal lines do not vary in the vertical direction.

Understanding horizontal lines helps simplify the process of graphing and analyzing linear equations.

Coordinate System

The coordinate system is a fundamental tool in mathematics for locating points on a plane. When graphing a linear equation like \(y = 8\), it's essential to understand this system. The most common is the Cartesian coordinate system, which consists of two number lines intersecting perpendicularly at a point called the origin (0,0).

• The horizontal line is called the x-axis.
• The vertical line is referred to as the y-axis.
• Each point on the plane is identified by an ordered pair \((x, y)\).

Once you understand the coordinate system, plotting any line, including the horizontal line \(y = 8\), becomes straightforward. You simply locate all the points where \(y\) is 8 while \(x\) can take any real number value from negative to positive infinity. This forms a line parallel to the x-axis.

Mastering the coordinate system is crucial because it lays the groundwork for more advanced graphing skills and analysis of geometric shapes.

Equation of a Line

The equation of a line provides a way to express a linear relationship between two variables, \(x\) and \(y\). In its simplest form, a line can be represented by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. However, for horizontal lines like \(y = 8\), the equation simplifies to \(y = c\), where \(c\) is a constant.

Here, some points to note about linear equations:

• The slope \(m\) describes the steepness and direction of a line. For horizontal lines, this slope \(m\) is zero.
• The y-intercept \(b\) tells us where the line crosses the y-axis. In the equation \(y=8\), this intercept is at 8.
• This equation allows us to calculate \(y\) for any given \(x\), solving for particular points on the line.

These points on a straight path formed by the equation of a line help us understand and visualize how one quantity varies with another in a consistent linear manner. Familiarity with these concepts serves as a vital skill in solving many mathematical and real-world problems.