Problem 29 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ y=-6 x-1 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -6, and the y-intercept is -1.

Step 1: Identify the slope of the line

The given equation is in the form of y = mx + b. Here, m represents the slope of the line. In our equation, y = -6x - 1, the slope (m) is -6.

Step 2: Identify the y-intercept of the line

In the slope-intercept form, y = mx + b, b represents the y-intercept. In our equation, y = -6x - 1, the y-intercept (b) is -1.

Step 3: Write the solution

The slope of the line with equation y = -6x - 1 is -6, and the y-intercept is -1.

Key Concepts

Understanding Slope

The slope is an essential concept when talking about linear equations. It is commonly denoted as \( m \) and describes the steepness and direction of a line. In mathematical terms, the slope is the ratio of the rise (change in \( y \)) over the run (change in \( x \)). Therefore, it shows how much \( y \) changes for a unit change in \( x \).

The slope can be positive, negative, zero, or undefined:

• A positive slope means the line rises as it moves from left to right.
• A negative slope indicates that the line falls as you move from left to right.
• A slope of zero means the line is horizontal; there is no change in \( y \).
• An undefined slope is associated with a vertical line where \( x \) does not change.

In our given equation, \( y = -6x - 1 \), the slope is -6, meaning the line falls steeply to the right. This tells us that for every unit increase in \( x \), \( y \) decreases by 6 units, which accounts for the negative slope direction.

Grasping the Y-Intercept

The y-intercept is a vital part of understanding a line on a graph. Represented by the \( b \) in the equation \( y = mx + b \), the y-intercept denotes the point where the line crosses the y-axis. At the y-intercept, the value of \( x \) is always zero.

Here's what that means for a graph:

• If the y-intercept is positive, the line crosses the y-axis above the origin.
• If it's negative, the line crosses below the origin.
• If the y-intercept is zero, the line passes through the origin (0,0).

In our case, with the equation \( y = -6x - 1 \), the y-intercept is -1. Thus, the line crosses the y-axis at the point (0, -1). This means the point on the y-axis that our line intersects is one unit below the origin.

Unraveling Linear Equations

Linear equations are the foundation for understanding straight lines on a graph. The classic form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Linear equations are characterized by:

• A constant rate of change, depicted through their uniform slope.
• Straight-line graphs, as they represent the simplest relationship between two variables.
• Being able to express relationships in many real-world scenarios like speed over time or cost versus quantity.

In our given problem, \( y = -6x - 1 \), we can rely on the equation's structure to easily identify both the slope and y-intercept as -6 and -1 respectively. Recognizing such equations allows one to quickly graph a line or determine how two variables interact with each other. By fully grasping these elements, you can unlock the powerful tool of linear relationships in various contexts.