Problem 69 - Calculus Volume 1

Question

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. $$ =\frac{1}{3}, \text { passes through }(0,4) $$

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{3}x + 4 \).

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a linear equation is written as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.

Step 2: Identify Slope and Y-Intercept

From the given conditions, the slope \( m \) is \( \frac{1}{3} \). The line passes through the point \((0,4)\), which means the y-intercept \( b \) is 4.

Step 3: Substitute Values into Slope-Intercept Form

Substitute \( m = \frac{1}{3} \) and \( b = 4 \) into the equation \( y = mx + b \). This gives \( y = \frac{1}{3}x + 4 \).

Key Concepts

Linear Equations

A linear equation represents a straight line on a graph and is fundamental in algebra and coordinate geometry. Often written in the form \( y = mx + b \), this equation captures all constant rate relationships between two variables. Here, \( y \) and \( x \) are variables which change based on the condition, \( m \) represents the slope, and \( b \) is the y-intercept. The primary characteristic of linear equations is that every change in \( x \) results in a predictable change in \( y \).
An easy way to visualize a linear equation is to look at it like a rule affecting two variables such that if you draw it on a graph, it will always create a straight line. No matter what values \( x \) takes (considering it is infinite), the equation remains faithful to its linear path.

Slope

The slope of a linear equation is a measure of its steepness and direction. It’s denoted as \( m \) in the equation \( y = mx + b \). Slope is calculated by the change in \( y \) divided by the change in \( x \), often summarized as \( \frac{\Delta y}{\Delta x} \).
When thinking about slope, consider these crucial points:

• A positive slope means the line goes upward as it moves from left to right. For example, \( m = \frac{1}{3} \) means for every 3 units moved horizontally, the line moves 1 unit upward.
• A negative slope indicates downward movement as the line extends from left to right.
• Zero slope results in a perfectly horizontal line, showing no vertical change no matter the horizontal movement.
• Lastly, an undefined slope arises in vertical lines where there's a zero horizontal change.

The slope is significant because it quantifies how one variable changes relative to another.

Y-Intercept

The y-intercept, represented as \( b \) in the slope-intercept equation \( y = mx + b \), is the point where the line crosses the y-axis. This intercept is crucial since it gives a starting value or pivot point for the line on a graph. When the x-value is zero, the y-value is the y-intercept.
For example, in the equation derived from the original problem, \( y = \frac{1}{3}x + 4 \), the y-intercept is 4. Meaning, when \( x \) is 0, \( y \) equals 4, which is exactly where the line meets the y-axis.
Understanding the y-intercept gives insight into the initial condition or position on a graph, making predictions about the behavior of \( y \) when we alter \( x \). It’s an accessible entry point for constructing or understanding the nature of the entire line.