Problem 60 - Algebra 2

Question

State the slope and the y-intercept of the graph of each equation. \(y=0.4 x+1.2\)

Step-by-Step Solution

Answer

Slope: 0.4, Y-intercept: 1.2

Step 1: Identify the Equation Form

The first step is to recognize that the equation given is in the slope-intercept form, which is: \[y = mx + b\]where \(m\) represents the slope, and \(b\) represents the y-intercept.

Step 2: Determine the Slope

In the equation \(y = 0.4x + 1.2\), the coefficient of \(x\) is \(0.4\). This value is the slope \(m\) of the line. Thus, the slope of the line is \(m = 0.4\).

Step 3: Determine the Y-intercept

The constant term in the equation \(y = 0.4x + 1.2\) is \(1.2\). This value is the y-intercept \(b\) of the line. Thus, the y-intercept of the line is \(b = 1.2\).

Key Concepts

Slope

The slope of a line is a measure of its steepness and direction. In any linear equation of the form \( y = mx + b \), the slope is represented by the coefficient \( m \). It's essentially telling you how much \( y \) increases or decreases as \( x \) increases by one unit. If you think of the line as a hill, the slope is how steep that hill is.

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A zero slope indicates a flat horizontal line.
• An undefined slope, often due to division by zero, represents a vertical line.

The slope provides critical insights into how two quantities correlate with each other. It's one of the first things you look for when analyzing the graph of a linear equation.

Y-intercept

The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the \( b \) represents the y-intercept. This is where the value of \( x \) is zero, and whatever value \( y \) attains at this point is your y-intercept.
Understanding the y-intercept helps you significantly in sketching the graph of linear equations and predicting the outcome when \( x \) is zero. It's a fixed value showing you where your line begins on the graph.
When interpreting data, the y-intercept can have different meanings depending on the context. In applied contexts, it often signifies the initial condition or starting point of the scenario being modeled. Recognizing the y-intercept is crucial for fully understanding the behavior of linear equations.

Linear Equations

Linear equations describe a constant relationship between two variables. They're called linear because their graph is a straight line. The general form of a linear equation is \( y = mx + b \). Understanding linear equations is fundamental to exploring many fields including science, engineering, and economics.

• They illustrate relationships with constant rates of change.
• They can model phenomena like speed, distance, or even population growth where the increase is constant.
• Their solutions form a line on a graph; hence the name "linear."

Linear equations help us predict values, understand trends, and compute unknowns in various situations. They are a stepping stone to more complex mathematics and thus crucial to master for any math student. When graphed, they give you a visual understanding of data trends over time or other varying conditions.