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Chapter 3: Problem 21

Find the slope and the -intercept of the line with the given equation. See Example 1 $$ y=\frac{x}{4}-\frac{1}{2} $$

Short Answer

Expert verified
Slope: \( \frac{1}{4} \), y-intercept: \( -\frac{1}{2} \).

Step by step solution

01

Understand the slope-intercept form

The equation of a line can be written in slope-intercept form, which is \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept.
02

Identify the slope and y-intercept

In the equation \( y = \frac{x}{4} - \frac{1}{2} \), compare it with the standard form \( y = mx + b \). Here, \( m = \frac{1}{4} \), which is the slope, and \( b = -\frac{1}{2} \), which is the y-intercept.
03

Conclusion

The slope of the line is \( \frac{1}{4} \) and the y-intercept is \( -\frac{1}{2} \).

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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 in algebra, presenting relationships between variables in a straight line when graphed on a coordinate plane. A linear equation is typically expressed in the form of \( y = mx + b \), known as the slope-intercept form. In this format:
  • \( y \) is the dependent variable, which changes with variations in \( x \).
  • \( m \) signifies the slope, indicating the steepness or incline of the line.
  • \( b \) stands as the y-intercept, representing where the line crosses the y-axis.
By comparing any given linear equation to this standard form, we can easily determine the slope and y-intercept, crucial for graphing or understanding the relationship between variables.
Exploring the Slope
The slope of a line, noted as \( m \) in the slope-intercept form, is a measure of how steep the line is. It represents the rate of change between the two variables:
  • When the slope is positive, as in \( m = \frac{1}{4} \), the line rises as it moves from left to right.
  • If the slope is negative, the line would fall as it moves from left to right.
  • A slope of zero indicates a horizontal line, implying no change as \( x \) changes.
In the equation \( y = \frac{x}{4} - \frac{1}{2} \), the slope \( \frac{1}{4} \) means that for every increase of 1 unit in \( x \), \( y \) increases by \( \frac{1}{4} \) unit. This results in a gently rising line.
Deciphering the Y-Intercept
The y-intercept, marked by \( b \) in the slope-intercept equation \( y = mx + b \), is the point where the line intersects the y-axis. This value is critical as it gives a starting point on the graph:
  • It represents the value of \( y \) when \( x \) is zero.
  • In context, it is the initial value or the constant term of a linear relationship.
  • For the equation \( y = \frac{x}{4} - \frac{1}{2} \), the y-intercept is \( -\frac{1}{2} \).
Thus, this point is \( (0, -\frac{1}{2}) \) on the graph, indicating when \( x \) is zero, \( y \) will be \( -\frac{1}{2} \). This understanding anchors the line on the graph, enabling a plot of points starting from the y-axis intersection.

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Most popular questions from this chapter

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \((0,0)\) and \((5,12)\)

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