Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(35-38\), find the slope and \(y\) -intercept of each line. \(3 y=-2 x+1\)

Step-by-Step Solution

Answer

Slope: \(-\frac{2}{3}\), Y-intercept: \(\frac{1}{3}\).

Step 1: Rearrange the Equation

The given equation is in the form of \[3y = -2x + 1\]To find the slope and y-intercept, we need to rearrange it into the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Divide each term by 3 to isolate \(y\):\[y = \frac{-2}{3}x + \frac{1}{3}\]

Step 2: Identify the Slope

The equation is now \[y = \frac{-2}{3}x + \frac{1}{3}\]In the slope-intercept form \(y = mx + b\), the slope \(m\) is the coefficient of \(x\). Hence, the slope of the line is \(-\frac{2}{3}\).

Step 3: Identify the Y-intercept

In the rearranged equation \[y = \frac{-2}{3}x + \frac{1}{3}\]the y-intercept \(b\) is the constant term. Therefore, the y-intercept of the line is \(\frac{1}{3}\).

Key Concepts

Understanding the Slope of a Line

The slope of a line is a measure of its steepness and direction. Imagine a hill; the slope indicates how steep it is. In mathematical terms, we denote the slope with the letter \(m\). Slope is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. It can be expressed as:

• Positive slope: The line rises from left to right.
• Negative slope: The line falls from left to right.
• Zero slope: The line is horizontal.
• Undefined slope: The line is vertical.

For the equation given in the exercise, once rearranged into slope-intercept form \(y = \frac{-2}{3}x + \frac{1}{3}\), the coefficient \(-\frac{2}{3}\) in front of \(x\) is our slope. Here, it is negative, meaning the line decreases from left to right. Understanding this concept helps you predict how the line will look without graphing it. For instance, a more negative slope makes the line steeper downward.

Role of Y-Intercept in Linear Equations

The y-intercept is the point where the line crosses the y-axis. This is the value of \(y\) when \(x\) equals zero. In the equation of a line in slope-intercept form \(y = mx + b\), the \(b\) represents the y-intercept. It is a pivotal aspect, as it tells us where the line starts in terms of height above or below the origin on the y-axis. In the given problem, after rearranging the equation, we get \(y = \frac{-2}{3}x + \frac{1}{3}\). The \(b\), or the y-intercept here, is \(\frac{1}{3}\). This implies that the line crosses the y-axis at the point \(\left(0, \frac{1}{3}\right)\).Knowing the y-intercept allows us to easily start graphing the line and gives us a clearer understanding of the line's placement on a Cartesian plane.

Exploring Linear Equations in Slope-Intercept Form

Linear equations are a fundamental aspect of algebra. They describe lines on a graph and can predict the relationship between two variables, like time and speed or cost and quantity. The slope-intercept form \(y = mx + b\) is particularly user-friendly because it clearly shows the slope and y-intercept.This form lets us quickly identify:

• The slope \(m\) of the line, indicating steepness and direction.
• The y-intercept \(b\), which shows where the line crosses the y-axis.

For example, in the equation \(y = \frac{-2}{3}x + \frac{1}{3}\), both the slope and y-intercept are easily extracted. The slope-intercept form is handy as it directly shows how changes in \(x\) impact \(y\), aiding in predicting outcomes and interpreting trends.