Problem 15 - Algebra A Combined Function

Question

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=-4, b=-\frac{1}{6} $$

Step-by-Step Solution

Answer

Equation: \( y = -4x - \frac{1}{6} \)

Step 1: Understand the Form of the Equation

The equation of a line in slope-intercept form is given by \[ y = mx + b \] where \( m \) is the slope of the line and \( b \) is the y-intercept. In our case, the slope \( m \) is given as \(-4\) and the y-intercept \( b \) is given as \(-\frac{1}{6}\).

Step 2: Substitute Slope and Y-Intercept

Substitute the given values of \( m \) and \( b \) into the slope-intercept formula. This gives us: \[ y = -4x - \frac{1}{6} \] This equation represents the line with the given slope and y-intercept.

Key Concepts

Slope

The slope of a line is a measure of its steepness or inclination. It represents how much the line rises or falls as it moves horizontally across a grid. In the context of a linear equation, the slope is denoted by the letter \( m \). To calculate the slope, you can use the formula \( m = \frac{\Delta y}{\Delta x} \), which represents the change in \( y \) over the change in \( x \).

• A positive slope means the line ascends from left to right.
• A negative slope indicates a decline from left to right.
• If the slope is zero, the line is horizontal.
• An undefined slope implies a vertical line.

Understanding the slope helps predict and graph the direction of the line, providing critical insight into its behavior in various mathematical and real-world applications.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. This is often represented as \( (0, b) \), where \( b \) is the value of the intercept. In an equation formatted in slope-intercept form, the y-intercept is the constant term, \( b \).

• This intercept provides information about the starting point of the line on the y-axis when \( x \) is zero.
• Knowing the y-intercept allows for easier graphing of the line because the intercept is one of the key anchor points.

For instance, in the exercise, the y-intercept is \( -\frac{1}{6} \), which tells us that the line crosses the y-axis at \( -\frac{1}{6} \). This starting point is crucial for drawing an accurate representation of the line.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. In mathematical terms, it is written as \( y = mx + b \). This form is particularly useful because it clearly shows both the slope \( m \) and the y-intercept \( b \). The slope-intercept form is ideal for quickly sketching the graph of a line and understanding its characteristics.

• The term \( mx \) indicates the direction and steepness of the line.
• The term \( b \) specifies where the line intersects the y-axis.

Using this form simplifies the process of graphing and analyzing linear relationships, making it a valuable tool in algebra and various applications.

Equation of a Line

An equation of a line mathematically represents a straight path defined by particular points in a plane. This equation encapsulates all the characteristics of the line, including its slope and y-intercept, when expressed in slope-intercept form \( y = mx + b \).
In our original exercise, the equation \( y = -4x - \frac{1}{6} \) describes a line with a slope of \( -4 \) and a y-intercept at \( -\frac{1}{6} \). This line portrays a downward sloping line, given the negative slope, crossing the y-axis slightly below zero. Such expressions are not just valuable in academics but also in fields such as physics, engineering, and economics where modeling linear relationships is essential.