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Chapter 2: Problem 21

For each equation, find the slope. If the slope is undefined, state this. $$ 5 y-4=35 $$

Short Answer

Expert verified
The slope is 0.

Step by step solution

01

- Isolate the variable

Rewrite the given equation in slope-intercept form, which is \( y = mx + b \). This makes it easier to identify the slope (m). Start by isolating y:5y - 4 = 35Add 4 to both sides:5y = 39
02

- Simplify the equation

Divide both sides of the equation by 5 to solve for y:\[ y = \frac{39}{5} \]
03

- Identify the slope

Compare your simplified equation \( y = \frac{39}{5} \) with the slope-intercept form \( y = mx + b \). Here, there is no x-term, so the equation represents a horizontal line where the slope (m) is 0.

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Key Concepts

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

slope-intercept form
To find the slope of a line from an equation, one of the most helpful steps is to rewrite the equation in slope-intercept form. The slope-intercept form is written as:

\( y = mx + b \)

Here, \( m \) represents the slope, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

This form is incredibly useful because it makes identifying the slope straightforward. Once you have the equation in this format, you can easily see what the slope \( m \) is by looking at the coefficient of the x-term. If there is no x-term, that already tells us something essential about the type of line it may represent.
isolating variable
When you are given an equation and need to find the slope, one of the first steps is to isolate the variable y. This means you need to rearrange the equation so that y is alone on one side of the equation.

Let's take the given example: \( 5y - 4 = 35 \).

1. Add 4 to both sides to start isolating y: \[ 5y - 4 + 4 = 35 + 4 \]
2. This simplifies to: \[ 5y = 39 \]
3. Next, divide both sides by the coefficient of y, which is 5: \[ y = \frac{39}{5} \]

By isolating y, we've rewritten the equation in a simpler form where y is by itself, making it easier to compare to the slope-intercept form.
horizontal line
A horizontal line is a special type of line in the context of slopes. In the equation derived above, i.e., \( y = \frac{39}{5} \), there is no x-term.

This indicates that the line is horizontal.

For any horizontal line, the slope (\( m \)) is always 0 because a horizontal line does not rise or fall; it remains constant.

Some characteristics of a horizontal line include:
  • All points on the line have the same y-coordinate.

  • The slope is 0, meaning there is no vertical change regardless of the horizontal movement.

Whenever you encounter an equation like \( y = b \), you can quickly conclude that it represents a horizontal line with a slope of 0.

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Most popular questions from this chapter

Find the indicated function values. $$ G(x)=\left\\{\begin{array}{ll}{x-5,} & {\text { if } x \leq-1} \\ {x,} & {\text { if } x>-1}\end{array}\right. $$ \(\begin{array}{lllll}{\text { a) } G(-10)} & {\text { b) } G(0)} & {} & {\text { c) } G(-1)}\end{array}\)

Indicate which of the following equations have \(\left(-\frac{1}{3}, \frac{1}{4}\right)\) as a solution. a) \(-\frac{3}{2} x-3 y=-\frac{1}{4}\) b) \(8 y-15 x=\frac{7}{2}\) c) \(0.16 y=-0.09 x+0.1\) d) \(2(-y+2)-\frac{1}{4}(3 x-1)=4\)

Find the intercepts. Then graph by using the intercepts, if possible, and a third point as a check. $$ 0.2 y-1.1 x=6.6 $$

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ 3 y=7(2 x-4) $$

Find the slope of the line that contains the given pair of points. a) \((5 b,-6 c),(b,-c)\) b) \((b, d),(b, d+e)\) c) \((c+f, a+d),(c-f,-a-d)\)
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