Problem 63 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. Slope \(=-11, \quad y\) -inter cept \(=-4\)

Step-by-Step Solution

Answer

Answer: The equation of the line is y = -11x - 4.

Step 1: Identify the slope and y-intercept

Given the slope \(m = -11\) and y-intercept \(b = -4\), we have the necessary information to write the equation of the line in slope-intercept form.

Step 2: Plug in the slope and y-intercept

We will plug in the values of slope and y-intercept into the slope-intercept form of the linear equation: \(y = mx + b\) Inserting given values: \(y = -11x - 4\) This is the equation of the line written in slope-intercept form.

Key Concepts

Linear Equations

Linear equations are fundamental mathematical expressions that describe a straight line. They are often presented in two main forms: the standard form and the slope-intercept form. Understanding these basic types helps in visualizing the relationship between variables and their graphical representation.

In general terms, a linear equation can be expressed as:

• Standard form: \( ax + by = c \)
• Slope-intercept form: \( y = mx + b \)

Each form has its unique advantages. The slope-intercept form is particularly useful for quickly identifying the slope and the y-intercept of the line.

These equations model linear relationships where one variable changes at a constant rate with respect to another. This constant rate of change is represented by the slope. Mastering linear equations allows for a deeper understanding of both algebra and real-world situations they can represent.

Slope

The slope of a line, often represented by the letter \(m\), serves as an indicator of the direction and steepness of that line. It plays a crucial role in understanding how the line behaves on a graph.

The slope can be defined in several ways depending on context:

• In mathematical terms, it's the ratio of the rise (change in y) over the run (change in x).
• From a geometric perspective, a positive slope indicates an upward slant from left to right, while a negative slope suggests a downward slant.
• A zero slope indicates a horizontal line, reflecting no change in y as x changes.
• An undefined slope typically corresponds to a vertical line, where changes in x have no impact on y.

In the slope-intercept form \( y = mx + b \), the slope directly influences the angle of the line, with steeper lines having larger absolute values.

Y-Intercept

The y-intercept is a key component in understanding where a line crosses the y-axis on a graph. Denoted by \(b\) in the slope-intercept form \(y = mx + b\), it provides important information about the start point of a line when \(x\) is zero.

Here’s how the y-intercept is significant:

• It is the value of \(y\) when \(x\) is zero. This means that in a graph, the line will touch the y-axis at this point.
• The y-intercept allows for a quick understanding of a line’s initial value, which can be particularly useful in data interpretation.
• In real-world terms, it can represent a starting quantity or an initial condition before any changes occur due to x.

Together with the slope, the y-intercept makes it easy to sketch a line and predict its behavior across different values of x.