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Chapter 2: Problem 16

For each equation, find the slope. If the slope is undefined, state this. $$ 19=-6 y $$

Short Answer

Expert verified
The slope is 0.

Step by step solution

01

Rewrite in slope-intercept form

The equation is given as \(19 = -6y\). Rewrite it to make it look like the slope-intercept form, \(y = mx + b\), where \(m\) represents the slope.
02

Solve for y

In order to get \(y\) by itself, divide both sides of the equation by \(-6\): \(\frac{19}{-6} = y\). Rearranging this, we get: \(y = -\frac{19}{6}\).
03

Identify the slope

Now that the equation is in the form \(y = mx + b\), we can see that \(y = -\frac{19}{6}\) has no \(x\) term. This means the slope \(m = 0\).

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Key Concepts

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

slope-intercept form
When working with linear equations, it's important to understand the slope-intercept form. The slope-intercept form of a linear equation is expressed as \[ y = mx + b \], where:
  • \textbf{y} is the dependent variable.
  • \textbf{x} is the independent variable.
  • \textbf{m} is the slope of the line.
  • \textbf{b} is the y-intercept, which is the point where the line crosses the y-axis.

The slope (\( m \)) indicates the steepness of the line and its direction.
If an equation like 19 = -6y is given, it can be rearranged into the slope-intercept form by solving for y.
This involves isolating y on one side of the equation:
\[-6y = 19 \] \[ y = -\frac{19}{6} \] Here, you can see that there is no x-term present, which simplifies our understanding of the line's slope.
slope of horizontal lines
A horizontal line has a special type of slope. Let's understand what happens:
When you see an equation like \[-\frac{19}{6} = y \], it translates to a horizontal line on the graph because the value of y is constant for all values of x.
A horizontal line means you're not going up or down as you move left or right across the graph.
The slope (\(m\)) for any horizontal line is always 0. Why? Because the rise (change in y) is 0 while the run (change in x) can be any number. Thus:
  • \[ m = \frac{\Delta y}{\Delta x} = \frac{0}{\Delta x} = 0 \]

In our exercise, when we rearranged the equation to get \[y = -\frac{19}{6} \], it essentially showed that y is always -\frac{19}{6}, no matter what value x takes, hence making the slope 0.
undefined slope
A slope can sometimes be undefined, which occurs in vertical lines.
For a vertical line, the equation looks like \[ x = c \], where \ (c) is a constant value. This means x never changes, but y can take any value.
The slope here is undefined because, in the formula for slope \ (m = \frac{\Delta y}{\Delta x}) \, the denominator (\( \Delta x \)) is 0, and division by zero is not defined.
  • \[ m = \frac{\Delta y}{0} \]
Since division by zero is undefined, so is the slope of a vertical line.
In summary:
  • Horizontal lines have a slope of 0. (\(y = c\))
  • Vertical lines have an undefined slope. (\(x = c\))
Understanding these concepts helps in easily identifying the slope and behavior of different types of lines.

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