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Chapter 3: Problem 19

Find the slope and the -intercept of the line with the given equation. See Example 1 $$ y=\frac{1}{2} x+6 $$

Short Answer

Expert verified
The slope is \( \frac{1}{2} \) and the y-intercept is \( 6 \).

Step by step solution

01

Identify the equation format

The equation of the line is given in the slope-intercept form, \[ y = mx + b \]where \( m \) represents the slope and \( b \) represents the y-intercept of the line.
02

Match equation components

Compare the given equation with the slope-intercept form. The given equation is \[ y = \frac{1}{2}x + 6 \]Here, \( \frac{1}{2} \) is analogous to \( m \), so it is the slope, and \( 6 \) is analogous to \( b \), so it is the y-intercept.
03

State the solution

From the comparison, we determine that the slope is \( \frac{1}{2} \) and the y-intercept is \( 6 \).

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Key Concepts

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

Understanding Slope
The slope is a fundamental concept in mathematics, especially when it comes to understanding linear equations. In the context of a line on a graph, the slope measures how much the line rises or falls as it moves from left to right. You can think of the slope as the "tilt" of the line.
  • The slope is often represented by the letter \( m \).
  • A positive slope (like \( \frac{1}{2} \) in our example) means the line is rising as it moves to the right.
  • A negative slope indicates the line is falling as it goes to the right.
  • If the slope is zero, the line is perfectly horizontal.
To calculate the slope from two points, you can use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). But in slope-intercept form equations like our example, you can directly "read" the slope as the coefficient of \( x \). It鈥檚 a powerful way to quickly assess the line's behavior.
The Role of the Y-intercept
The y-intercept is the point where the line crosses the y-axis of a graph. This is another crucial component when analyzing linear equations. In simpler terms, it tells you where the line will start if it begins on the y-axis.
  • The y-intercept is denoted by the letter \( b \).
  • In our example, the y-intercept is 6. This means that when \( x = 0 \), \( y \) is 6.
  • The y-intercept provides a starting point for the line and is essential for sketching it on a graph.
To find the y-intercept in a slope-intercept form equation, look for the constant term. This term is always added or subtracted independently and typically follows the slope component.
Linear Equations Demystified
Linear equations form the foundation of algebra and are vital for analyzing relationships between variables. A linear equation describes a straight line and is commonly written in the slope-intercept form: \( y = mx + b \).
  • Linear equations create straight lines on a graph, without any curves.
  • They are called "linear" because the graph of their solutions forms a line.
  • The slope and y-intercept define the unique characteristics of each line.
Understanding linear equations helps in solving real-world problems where relationships between quantities are constant, like predicting expenses over time or determining speed. By recognizing the slope and y-intercept, you can form a complete picture of how these equations work and how they visually appear on a graph.

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Most popular questions from this chapter

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