Problem 17 - Algebra 1

Question

Find the slope and the y-intercept of the graph of the equation. $$ y=-2 $$

Step-by-Step Solution

Answer

The slope of the line is 0 and the y-intercept is -2.

Step 1: Identifying the Form of Equation

The given equation 'y=-2' is in the form of \(y=mx+c\) where \(m\) is the slope and \(c\) is the y-intercept. Here, the equation 'y=-2' can be rewritten as 'y=0x-2'.

Step 2: Identifying the Slope

In the equation \(y=0x-2\), the coefficient ofx is 0. So, the slope of the line is 0.

Step 3: Identifying the y-Intercept

In the equation \(y=0x-2\), the constant term is -2. So, the y-intercept of the line is -2.

Key Concepts

Linear Equations

Linear equations are fundamental components in algebra and help us describe a straight line graphically. These equations typically have the form \(y = mx + c\), where:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\) is the slope of the line, which shows the steepness and direction.
• \(c\) is the y-intercept, or where the line crosses the y-axis.

Linear equations can represent a diverse range of scenarios like constant rates and relationships between quantities.
For example, the equation \(y = -2\) is a special type of linear equation.
Here, it can be seen as \(y = 0x - 2\) meaning that no matter the value of \(x\), \(y\) will always be -2. This results in a horizontal line across the y-axis at -2, a characteristic of zero slope equations.

Slope

The slope of a line tells us how steep the line is and its direction - whether it's rising, falling, or flat.
In mathematical terms, the slope is represented by the letter \(m\) in the equation \(y = mx + c\).
The slope can often be calculated as the ratio of the vertical change to the horizontal change between two points on the line.

• A positive slope means the line rises as it moves from left to right.
• A negative slope indicates the line falls.
• A slope of zero means the line is perfectly horizontal.

In the given equation \(y = -2\), rewritten as \(y = 0x - 2\), the coefficient of \(x\) (which is \(0\)) represents the slope.
This tells us that the line doesn't rise or fall, making it a horizontal line.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis.
In the equation \(y = mx + c\), the constant \(c\) denotes this intercept.
The y-intercept is helpful for quickly drawing graphs without needing to plot multiple points.

• If you know the slope and the y-intercept, you can determine the line's direction and position on the graph.
• It is essentially the value of \(y\) when \(x\) equals zero.

For the equation \(y = -2\), seen further as \(y = 0x - 2\), the constant term is \(-2\).
This indicates that the line crosses the y-axis at -2.
This value helps affirm that the line is indeed horizontal, since there's no change along the x-axis regardless of its value.