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Chapter 2: Problem 36

For each equation, find the slope. If the slope is undefined, state this. $$ 2 x=10 $$

Short Answer

Expert verified
The slope is undefined.

Step by step solution

01

Identify the given equation

The given equation is: \( 2x = 10 \)
02

Solve for x

Divide both sides of the equation by 2 to solve for \( x \): \( x = \frac{10}{2} \)\( x = 5 \)
03

Rewrite the equation in slope-intercept form

The equation \( x = 5 \) represents a vertical line. A vertical line cannot be written in the slope-intercept form \( y = mx + b \), where \( m \) is the slope.
04

Determine the slope

Since the equation represents a vertical line, the slope is undefined.

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Key Concepts

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

Vertical Line
A vertical line is a straight line that goes up and down parallel to the y-axis. In the coordinate system, it will always have an equation in the form of \( x = a \), where \( a \) is a constant.

Unlike other lines, a vertical line does not depend on the \( y \) coordinate, meaning its \( x \) coordinate will always be the same no matter what the value of \( y \) is.

For instance, the given equation \( 2x = 10 \) simplifies to \( x = 5 \), which represents a vertical line passing through \( x = 5 \).
  • Vertical lines are parallel to the y-axis.
  • All points on the line have the same x-coordinate.
  • Vertical lines are not functions because they do not pass the vertical line test (a single vertical line cannot intersect with a vertical line at more than one point).
Undefined Slope
The 'slope' of a line measures how steep the line is. Typically, the slope \( m \) is calculated using the formula \[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \].

However, for vertical lines, the difference in the x-coordinates will always be zero since all points on a vertical line have the same x-coordinates.

If you attempt to calculate the slope of a vertical line, you end up with division by zero: \[ m = \frac{(y_2 - y_1)}{0} \]

Division by zero is undefined in mathematics, so the slope of a vertical line is considered to be undefined.
  • The slope of vertical lines is always undefined.
  • They are neither positive nor negative.
  • An undefined slope makes it impossible to write the equation in slope-intercept form.
Solving Linear Equations
Solving linear equations involves finding the value of the variable that makes the equation true. For the given equation, \(2x = 10\), the process was as follows:
  • Step 1: Identify the equation. The given equation is \( 2x = 10 \).
  • Step 2: Solve for the variable. Divide both sides by 2 to isolate \( x \):
    \(\frac{2x}{2} = \frac{10}{2}\)
    So, \( x = 5 \).
  • Step 3: Rewrite in appropriate form. Recognize that the equation \( x = 5 \) represents a vertical line.

    Since vertical lines cannot be represented in slope-intercept form, the solution ends here.
Efficiently solving linear equations is a crucial skill in algebra, and understanding that certain forms like vertical lines have unique properties helps in mastering this skill.

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Most popular questions from this chapter

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