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Chapter 2: Problem 24

For each equation, find the slope. If the slope is undefined, state this. $$ x-4 y=12-4 y $$

Short Answer

Expert verified
The slope is undefined.

Step by step solution

01

- Simplify the Equation

First, we want to simplify the given equation. Start by subtracting \(-4y\) from both sides:\[x - 4y = 12 - 4y\]Simplifying this gives:\[x = 12\]
02

- Understand the Simplified Equation

Now observe the simplified equation: \(x = 12\). This represents a vertical line where the value of x is always 12, regardless of the value of y.
03

- Determine the Slope

Recall that the slope of a vertical line is undefined, because it does not change in the y-direction. Hence, the slope of \(x = 12\) is undefined.

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Key Concepts

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

Solving Equations
When solving equations, our goal is to isolate the variable on one side of the equation. In this exercise, the equation is given by:
\(x - 4y = 12 - 4y\).
To simplify, we subtract \(-4y\) from both sides:
$$x - 4y + 4y = 12 - 4y + 4y$$
This simplifies to:
$$x = 12$$
Achieving this simplification helps us understand the nature of the equation better.
Remember, we always perform the same operation on both sides of the equation to maintain equality.
If you ever get stuck, try reversing the operations step by step.
Vertical Lines
Once simplified, our equation becomes:
$$x = 12$$.
This type of equation represents a vertical line.
In a Cartesian coordinate system, a vertical line means that for every point on that line, the x-coordinate remains constant, which in this case is 12.
The y-coordinate, however, can take any value.
For example, the points (12, 1), (12, -5), and (12, 100) all lie on the line \(x = 12\). Vertical lines are unique when compared to other lines because they run parallel to the y-axis and do not tilt in any direction.
Slope Determination
The slope of a line indicates how steep the line is.
To find the slope, we use the formula:
$$\text{slope} = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1}$$.
For vertical lines, there is no change in the x-coordinate; \(x_1 = x_2\).
This makes the denominator zero, leading to:
$$\frac{\text{change in y}}{0}$$
Division by zero is undefined, therefore, vertical lines have an undefined slope.
If you see an equation of the form \(x = a\) (where 'a' is a constant), it's always a vertical line with an undefined slope.
Occurs because the line does not rise or fall鈥攊t just moves vertically.

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Most popular questions from this chapter

Explain the difference between finding \(f(0)\) and finding the input \(x\) for which \(f(x)=0\)

Using the window \([-5,5,-1,9],\) graph \(y_{1}=5\) \(y_{2}=x+2,\) and \(y_{3}=\sqrt{x} .\) Then predict what shape the graphs of \(y_{1}+y_{2}, y_{1}+y_{3},\) and \(y_{2}+y_{3}\) will take. Use a graphing calculator to check each prediction.

Solve each equation graphically. Then check your answer by solving the same equation algebraically. $$ 3 x+7=4 $$

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ 8 x+40=0 $$

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