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Chapter 2: Problem 17

Determine the \(y\) -intercept. $$ y=-\frac{3}{8} x-4.5 $$

Short Answer

Expert verified
The y-intercept is \(-4.5\).

Step by step solution

01

Identify the equation format

The equation given is in the slope-intercept form, which is written as \( y = mx + b \). In this format, \( m \) represents the slope and \( b \) represents the y-intercept.
02

Locate the y-intercept

From the equation \( y = -\frac{3}{8} x - 4.5 \), compare it to the standard form \( y = mx + b \). Here, \( b = -4.5 \), which indicates the y-intercept.
03

State the y-intercept

The y-intercept of the given equation is \( -4.5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
One of the most common ways to write a linear equation is by using the slope-intercept form. The slope-intercept form of a linear equation is written as
\( y = mx + b \).
  • \( m \): This represents the slope of the line. The slope is how steep the line is and the direction it goes in. It is calculated as the change in \( y \) (vertical) over the change in \( x \) (horizontal).
  • \( b \): This is the y-intercept. It is the point where the line crosses the y-axis.

In the given exercise, the equation is \( y = -\frac{3}{8} x - 4.5 \). Here, the slope \( m \) is -\(\frac{3}{8} \) and the y-intercept \( b \) is -4.5. This form helps easily identify these two key components.
linear equations
Linear equations are mathematical expressions that describe straight lines. They are useful because they represent a constant rate of change. A linear equation can be written in various forms, but the slope-intercept form is one of the most straightforward.
Key characteristics include:
  • Constant Slope: The slope of a linear equation remains constant no matter where you measure it on the line.
  • One-to-One Relationship: Each value of \( x \) has one and only one corresponding value of \( y \), making it a function.
  • Straight Line: Graphing a linear equation results in a straight line.

For example, the given exercise features the linear equation \( y = -\frac{3}{8} x - 4.5 \), which describes a straight line with a constant slope of -\(\frac{3}{8} \) and a y-intercept of -4.5.
y-intercept
The y-intercept is a fundamental concept when dealing with linear equations. It is the point where the line crosses the y-axis, which means the value of \( x \) is zero at this point.

To find the y-intercept in the slope-intercept form \( y = mx + b \), you simply look at the value of \( b \). In the given exercise \( y = -\frac{3}{8} x - 4.5 \), the y-intercept is -4.5.

Understanding the y-intercept helps in:
  • Graphing: It provides a starting point for graphing the line on a coordinate plane.
  • Equations: It offers insight into the relationship between \( x \) and \( y \) in the equation.
  • Analysis: It allows quick identification of a specific point through which the line passes.

For any linear equation, the y-intercept is crucial for understanding and graphing the line effectively.

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