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Chapter 2: Problem 14

For each equation, find the slope. If the slope is undefined, state this. $$ y-3=5 $$

Short Answer

Expert verified
The slope is 0.

Step by step solution

01

Identify the given equation

The given equation is \( y - 3 = 5 \).
02

Simplify the equation

To simplify the equation, solve for \( y \). Add 3 to both sides: \( y - 3 + 3 = 5 + 3 \) which simplifies to \( y = 8 \).
03

Interpret the simplified equation

The simplified equation \( y = 8 \) represents a horizontal line.
04

Determine the slope

The slope of a horizontal line is 0 because there is no change in the \( y \)-value as the \( x \)-value changes.

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Key Concepts

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

Horizontal Line
A horizontal line is a straight line that runs left to right across the coordinate plane. It has the unique property where its slope is always zero. This is because, for any two points on a horizontal line, the y-coordinate remains constant while the x-coordinate varies as it moves horizontally. In simpler terms, there's no rise when you move from one point to another on that line, only a run. For example, the equation we simplified, \( y = 8 \), tells us that all points on this line have a y-value equal to 8, no matter what the x-value is. Therefore, the line is horizontal.
Solving Equations
Solving equations is the process of finding the values of variables that satisfy the given mathematical statements. For the equation given in the exercise, \( y - 3 = 5 \), the goal is to isolate the variable \( y \). To do this, we perform operations that simplify the equation. Here are the steps we followed:
  • Add 3 to both sides to isolate \( y \): \( y - 3 + 3 = 5 + 3 \)
  • This simplifies to: \( y = 8 \)
The simplified equation \( y = 8 \) is now in its simplest form, making it easier to interpret and use.
Slope Calculation
The slope of a line indicates its steepness and direction, calculated by the change in the y-values divided by the change in the x-values. For the equation \( y = 8 \), representing a horizontal line, we can understand this concept better. On a horizontal line, all y-values are the same, meaning the difference in y-values (rise) is zero no matter how far apart the x-values (run) are. Thus, the slope formula \( m = \Delta y / \Delta x \) simplifies as: \ \ \[ m = \frac{0}{\Delta x} = 0 \] This confirms that the slope of any horizontal line is zero. It's essential to remember that while vertical lines have undefined slopes, horizontal lines always have a slope of zero.

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