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Chapter 4: Problem 4

What is the \(y\) -intercept and slope of the line with the equation \(y=4 x+\) \(3 ?\) (A) y-intercept: 4 and slope: 3 OR (B) y-intercept: 3 and slope: 4

Short Answer

Expert verified
The y-intercept is 3 and the slope is 4. Option B is correct.

Step by step solution

01

Identify the Equation Format

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

Extract the Slope

In the equation \( y = 4x + 3 \), compare it to \( y = mx + b \). Here, \( m = 4 \), which means the slope of the line is 4.
03

Identify the Y-Intercept

Continuing from the equation \( y = 4x + 3 \), the constant term \( b = 3 \), indicating that the y-intercept of the line is 3.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way of writing the equation of a line so that it is easy to read off the slope and the y-intercept. It is represented as \( y = mx + b \). In this formula, \( y \) and \( x \) are variables that represent any point on the line, while \( m \) indicates the slope of the line, and \( b \) specifies the y-intercept.
The main advantage of using slope-intercept form is that it provides a quick glance at two crucial components of the line's equation without further calculation.
  • Fast identification of line characteristics.
  • Easy to plot on a graph.
  • Helps to predict the behavior of the line.
Knowing the slope-intercept form makes it straightforward to analyze line equations and carry out graph-related tasks.
Slope
The slope of a line refers to how steep the line is, and is represented by the letter \( m \) in the slope-intercept formula. In the equation \( y = 4x + 3 \), the slope \( m \) is 4.
Slope indicates the direction and the steepness of a line:
  • If the slope is positive, the line inclines upwards as we move from left to right.
  • A negative slope would indicate a downward incline.
  • A slope of 0 signifies a perfectly horizontal line.
The slope is calculated by the ratio of the vertical change, or "rise," over the horizontal change, or "run" (\( m = \frac{\text{rise}}{\text{run}} \)). This key characteristic allows you to determine how sharply a line ascends or descends.
Y-Intercept
The y-intercept is where the line crosses the y-axis, and it is represented by the constant \( b \) in the slope-intercept form \( y = mx + b \). For the equation \( y = 4x + 3 \), the y-intercept \( b \) is 3.
This point occurs at \((0, b)\), meaning it is the y-value when \( x \) is 0:
  • The y-intercept is a key starting point for graphing a line.
  • It provides the initial value of \( y \).
  • Helps to quickly establish the line's positioning on a graph.
Knowing the y-intercept makes it much easier to start plotting the line, as it gives you a specific point to place right on the graph's y-axis.

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