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Chapter 3: Problem 83

In the following exercises, find the slope of each line. $$ x=-5 $$

Short Answer

Expert verified
The slope is undefined.

Step by step solution

01

Identify the Type of Line

Recognize that the equation of the line given is in the form of \(x = c\), where \(c\) is a constant. This represents a vertical line.
02

Understand Vertical Line Properties

Recall that a vertical line goes straight up and down, and therefore does not have a defined slope.
03

State the Slope of the Line

Since vertical lines do not have a defined slope, the slope of the line \(x = -5\) is undefined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Lines
Vertical lines are unique and easy to identify on a graph. They run straight up and down and are defined by the equation of the form \(x = c\), where \(c\) is a constant number. This means that no matter what the value of \(y\) is, the value of \(x\) will always be the same. For example, in the equation \(x = -5\), \(x\) will always be -5 regardless of the \(y\) value. Vertical lines are distinct from horizontal lines, which run left to right and are represented by equations like \(y = c\).
Vertical lines play a critical role in understanding geometry and algebra.
Undefined Slope
The concept of slope is essential in graphing linear equations. Slope measures the steepness or incline of a line and is usually represented by the letter \(m\). For most lines, slope is defined as the 'rise over run,' which means the vertical change divided by the horizontal change between two points on the line. However, vertical lines are different.
In the case of vertical lines, such as \(x = -5\), there is an infinite vertical change but zero horizontal change. This translates mathematically to a division by zero situation, which is undefined. Consequently, vertical lines do not have a slope; we say their slope is 'undefined.'
Linear Equations
Linear equations represent straight lines when plotted on a graph. There are various ways these equations can be written, but one of the most common forms is the slope-intercept form \(y = mx + b\). In this equation, \(m\) represents the slope, and \(b\) represents the y-intercept, the point where the line crosses the y-axis.
Linear equations can also be written in standard form \(Ax + By = C\), where \(A, B,\) and \(C\) are constants. In the case of vertical lines, the equation is simplified to the form \(x = c\), focusing solely on the \(x\) value. Understanding the different forms of linear equations helps in graphing and solving algebraic problems.

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Most popular questions from this chapter

In the following exercises, use the set of ordered pairs to ( ) determine whether the relation is a function, (b) find the domain of the relation, and \(\odot\) find the range of the relation. \(\\{(-3,27),(-2,8),(-1,1),\) (0,0),(1,1),(2,8),(3,27)\\}

Graph the linear inequality: \(y>-2 x+1\)

In the following exercises, determine whether each equation is a function.. (1) \(2 x-4 y=8\) (b) \(-4=x^{2}-y\) (c) \(y^{2}=-x+5\)

In the following exercises, use the set of ordered pairs to ( ) determine whether the relation is a function, (b) find the domain of the relation, and \(\odot\) find the range of the relation. \(\\{(-3,-27),(-2,-8),(-1,-1),\) (0,0),(1,1),(2,8),(3,27)\\}

In the following exercises, evaluate the function. $$ h(t)=2|t-5|+4 ; h(-4) $$
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