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Chapter 0: Problem 13

Graph. List the slope and y-intercept. $$ y=3 x-4 $$

Short Answer

Expert verified
The slope is 3, and the y-intercept is -4.

Step by step solution

01

Identify the Equation Form

The given equation is \( y = 3x - 4 \). This is written in the slope-intercept form of a line, which is \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.
02

Determine the Slope

In the equation \( y = 3x - 4 \), compare it with \( y = mx + b \). The coefficient of \( x \), which is 3, represents the slope \( m \). Thus, the slope of the line is 3.
03

Find the Y-Intercept

In the equation \( y = 3x - 4 \), the constant term \(-4\) represents the y-intercept \( b \). This means the y-intercept of the line is -4.

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Key Concepts

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

Understanding Linear Equations
Linear equations are fundamental to algebra and represent straight lines when graphed on a coordinate plane. The most common form is the slope-intercept form, expressed as \( y = mx + b \). In this form:
  • \( y \) and \( x \) are variables representing coordinates on a graph.
  • \( m \) is the slope of the line, which describes the line's steepness and direction.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
The slope-intercept form makes it easy to quickly sketch the graph of a linear equation by simply identifying these two key components.
Decoding the Slope
The slope of a linear equation is a crucial component that tells us how the line inclines or declines. It is often denoted by \( m \) in the equation \( y = mx + b \). In this context:
  • The slope is the change in the vertical direction (up/down) for each unit of change in the horizontal direction (left/right).
  • For example, a slope of 3 means that for each step you move to the right, the line goes up by 3 units.
  • A positive slope indicates an upward trend from left to right, while a negative slope indicates a downward trend.
In the equation \( y = 3x - 4 \), the slope \( m \) is 3, indicating a rising line that ascends 3 units for each unit it moves horizontally.
Exploring the Y-Intercept
The y-intercept is where the line crosses the y-axis on a graph. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). Understanding the y-intercept helps in:
  • Identifying the point where the line starts on the y-axis when \( x = 0 \).
  • Making it easier to graph the line by providing a starting point for drawing the line.
  • Comparing lines to see how their positions differ on the graph.
In the equation \( y = 3x - 4 \), the y-intercept is -4. This indicates that when the line crosses the y-axis, it is at the point (0, -4). This point serves as a crucial anchor for plotting the line accurately.

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