Problem 70 - Algebra 1

Question

If \(y+8=0,\) what is the slope of the graph? $$ \begin{array}{lllll} \text { (A) Undefined } & \text { (B) } 1 & \text { (C) } 0 & \text { (D) }-1 \end{array} $$

Step-by-Step Solution

Answer

The slope of the graph is 0. So, the correct answer is (C) 0.

Step 1: Identifying the form of the equation

The given equation is \(y + 8 = 0\). This equation can be rewritten as \(y = -8\) which matches the format of a linear equation \(y = mx + c\) where \(m\) is the slope and \(c\) is the y-intercept.

Step 2: Recognizing the lack of x component

In the rewritten equation \(y = -8\), there is no \(x\) component. This means that for every \(x\), the \(y\) value does not change, it remains -8. This tells us that the line is horizontal.

Step 3: Determining the slope

The slope of a line is a measure of the vertical change (the change in y-values) for each unit of horizontal change (the change in x-values). For a horizontal line, there is no vertical change regardless of the horizontal change, meaning the slope of a horizontal line is 0.

Key Concepts

Linear Equations

Linear equations are at the heart of algebra and serve as a foundational concept in understanding mathematical relationships. They are equations that describe a straight line on a graph. A standard form of a linear equation is \(y = mx + c\), where:

• \(m\) is the slope, representing the steepness of the line or the rate at which \(y\) changes with respect to \(x\).
• \(c\) is the y-intercept, indicating the point where the line crosses the y-axis.

In this exercise, we encountered the equation \(y + 8 = 0\), which we rewrote as \(y = -8\). This is a special case of a linear equation, known as a horizontal line, as it lacks an \(x\) term and means that the equation won't vary with changes in \(x\). The slope of the equation is crucial as it determines how the line behaves, whether it rises, falls, or remains constant across the graph.

Horizontal Line

A horizontal line is one of the simplest forms of linear equations and can be easily recognized as it runs straight across a graph parallel to the x-axis. The equation \(y = -8\) describes a horizontal line, because:

• The value of \(y\) remains constant (-8 in this case) for any given \(x\).
• This constant value results in no tilt or slope to the line, distinguishing it from diagonal or vertical lines.
• Graphically, it appears as a flat line situated at \(y = -8\) across the entirety of the x-axis.

Understanding the concept of a horizontal line helps in identifying how slopes function, especially since the slope for all horizontal lines is always zero. This feature makes horizontal lines important in contrasting different linear relationships based on the slopes that define how dynamic or static the relationships are.

Slope Calculation

Slope calculation is a fundamental aspect of graphing and interpreting linear equations. The slope \(m\) is computed as the "rise" over the "run," or the change in \(y\) over the change in \(x\):\[m = \frac{\Delta y}{\Delta x}\]For horizontal lines, such as \(y = -8\), there is no change in the \(y\) value as \(x\) changes. Thus, \(\Delta y = 0\). This means that, for all movements along the x-axis, the vertical height (rise) of the line remains unchanged. Consequently, the slope of horizontal lines is always zero:

• \(m = \frac{0}{\Delta x} = 0\)
• This zero slope signifies a total lack of vertical inclination.
• The graph maintains a uniform horizontal course across its extent.

Mastering slope calculation is vital for students to analyze how lines relate to each other, allowing them to identify trends, project data, and solve problems that hinge on understanding linear relationships.