Problem 21 - Algebra A Combined Function

Question

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=-\frac{1}{5}, b=\frac{1}{9} $$

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{1}{5}x + \frac{1}{9} \).

Step 1: Understanding the Slope-Intercept Form

To write the equation of a line given a slope and a y-intercept, we use the slope-intercept form of a linear equation. This form is expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Substitute the Given Slope and Intercept

Take the given slope \( m = -\frac{1}{5} \) and the y-intercept \( b = \frac{1}{9} \). Substitute these values into the slope-intercept form, replacing \( m \) with \( -\frac{1}{5} \) and \( b \) with \( \frac{1}{9} \).

Step 3: Write the Final Equation

After substitution, the equation becomes \( y = -\frac{1}{5}x + \frac{1}{9} \). This is the equation of the line with the given slope and y-intercept.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing the equation of a straight line. It's very popular in the realm of linear equations due to its simplicity and clarity. This form is written as:

• \( y = mx + b \)

Here, \( y \) and \( x \) are the coordinates of any point on the line.
\( m \) represents the 'slope' of the line, which tells us how steep the line is.
\( b \) is the 'y-intercept,' which specifies the point where the line crosses the y-axis.
The slope-intercept form is particularly useful because it allows you to easily read off both the slope and the y-intercept, enabling a quick sketch or interpretation of the line’s behavior.
This equation makes it easy to visualize how the line changes as it moves across the graph.

Slope

In the context of linear equations, the slope describes how steep a line is. It's essentially a measure of the line's gradient or incline. The mathematical representation is as follows:

• \( m = \text{rise over run} = \frac{\Delta y}{\Delta x} \)

This formula means you take the change in \( y \) (vertical movement) and divide it by the change in \( x \) (horizontal movement).
A positive slope indicates the line is ascending as it moves from left to right across the graph.
Conversely, a negative slope, like \(-\frac{1}{5}\), means the line is descending.
The more substantial the slope, the steeper the line. A slope of zero means the line is perfectly horizontal, while an undefined slope (division by zero) indicates a vertical line.
Understanding the slope is crucial because it affects many characteristics of the graph that represents the linear equation.

Y-Intercept

The y-intercept is a key feature of a line in the coordinate plane. It gives you a starting point to draw it visibly on a graph. The y-intercept is defined as the specific point where the line crosses the y-axis.

• In the slope-intercept equation form \( y = mx + b \), the term \( b \) represents the y-intercept.

To find this point, set \( x = 0 \) in the equation. The resulting \( y \) value is your y-intercept.
For instance, in the equation \( y = -\frac{1}{5}x + \frac{1}{9} \), the y-intercept is \( \frac{1}{9} \).
This point is useful because it provides an anchor from which the line starts moving in the direction described by the slope.
Graphically understanding this point helps in plotting and interpreting the behavior of linear equations.