Problem 19 - Algebra A Combined Function

Question

Mixed Practice Find the slope of each line. See Examples 3 through 6. $$ y=-0.3 x+2.5 $$

Step-by-Step Solution

Answer

The slope of the line is -0.3.

Step 1: Identify the Slope-Intercept Form

The equation given is in slope-intercept form, which is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Compare the Equation with the Slope-Intercept Form

Compare the given equation \(y = -0.3x + 2.5\) with the general slope-intercept form \(y = mx + b\). Here, \(m\) is the coefficient of \(x\), which is \(-0.3\).

Step 3: Identify the Slope

The slope \(m\) can be directly taken from the equation, which is \(-0.3\). This is the coefficient of \(x\) in the equation \(y = -0.3x + 2.5\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most widely used because it provides clear insight into the properties of the line it represents. A linear equation in this form is expressed as \( y = mx + b \). Here, \( m \) stands for the slope of the line, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

This form is particularly useful because it allows one to easily identify the slope and y-intercept directly from the equation. This makes graphing or understanding the behavior of the line much more straightforward. The slope-intercept form shows how the value of \( y \) changes with respect to \( x \).

• By looking at \( m \), you can determine whether the line rises or falls.
• The value of \( b \) tells you where the line crosses the y-axis, which is crucial when graphing.

Linear Equations

Linear equations are at the heart of algebra. These are equations that describe straight lines when graphed on the coordinate plane. The general form of a linear equation is \( ax + by = c \), but it can also be rearranged into slope-intercept form, \( y = mx + b \).

These equations are called 'linear' because they graph as straight lines. No matter the complexity of real-world problems, if they can be represented using linear equations, the relationships and interactions are usually simple and easy to analyze:

• The solutions to linear equations represent points on a line.
• They are characterized by a constant rate of change, shown as the slope \( m \).

Understanding linear equations is crucial because they serve as a foundation for more advanced math topics. They allow students to model relationships between variables in problems across various contexts.

Coefficient of x

In the context of linear equations, the coefficient of \( x \) plays a pivotal role. It is the number in front of the \( x \) term in the equation and directly relates to the slope of the line. For instance, in the equation \( y = -0.3x + 2.5 \), the coefficient of \( x \) is \(-0.3\).

The coefficient of \( x \) not only indicates the steepness of the line but also its direction:

• If the coefficient is positive, the line slopes upwards from left to right.
• If it's negative, like \(-0.3\), the line slopes downwards.

This is a vital concept because it links abstract numbers to visual understanding. By merely looking at the coefficient of \( x \), one can quickly tell how the line behaves on a graph, which is invaluable for both solving equations and interpreting data.