Problem 17 - Calculus: One and Several Variables

Question

Find the slope and \(y\) -intercept. $$3 y=x+6$$

Step-by-Step Solution

Answer

The slope of the line is 1/3 and the y-intercept is 2.

Step 1: Rewrite the equation in slope-intercept form

To get the slope and the y-intercept, the equation must be in the form of y = mx + b. Currently the equation is 3y = x + 6. Therefore, let's divide all terms by 3. This results in y = 1/3x + 2.

Step 2: Identify the slope

Now that the equation is in the proper form, it can be seen that the coefficient of x is the slope (m). Therefore, in the equation y = 1/3x + 2, the slope is 1/3.

Step 3: Identify the y-intercept

The constant term in the equation is the y-intercept (b). Therefore, in the equation y = 1/3x + 2, the y-intercept is 2.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra. In general, a linear equation is an equation that forms a straight line when graphed on a coordinate plane. This is why they are called "linear." The simplest form to represent a linear equation is the slope-intercept form, written as \( y = mx + b \). Here:

• \( y \) is the dependent variable, often representing an output or result.
• \( x \) is the independent variable, which could be any given input.
• \( m \) represents the slope, which indicates the steepness or inclination of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Changing the values of \( m \) and \( b \) in the equation will alter the line's appearance and position on a graph. Linear equations are useful for modeling situations where there is a constant rate of change, like speed or growth over time. Understanding how to manipulate these equations helps solve various real-world problems.

Slope

The slope of a line is a measure of its steepness and direction. In the equation of a line \( y = mx + b \), the slope is the coefficient \( m \). It tells us how much \( y \) increases or decreases as \( x \) increases by one unit. Thinking about slope can be simplified as "rise over run," referring to how much change occurs in the vertical direction (rise) per unit of change in the horizontal direction (run).

• If \( m \) is positive, the line rises as it moves from left to right.
• If \( m \) is negative, the line falls as it moves from left to right.
• If \( m = 0 \), the line is horizontal, indicating no change in \( y \) regardless of \( x \).

The slope can also reveal how variables in a real-world scenario are related, like how the number of hours worked (\( x \)) might affect total earnings (\( y \)). A steeper slope means a greater increase or decrease in \( y \) for each increase in \( x \). In our example, the slope is \( \frac{1}{3} \), meaning for every unit increase in \( x \), \( y \) increases by \( \frac{1}{3} \).

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis. In the slope-intercept form \( y = mx + b \), the \( b \) represents this y-intercept. It is the value of \( y \) when \( x = 0 \). Such a point on a graph is straightforward to identify because one just needs to look at where the line meets the y-axis.

• If \( b \) is a positive number, the line crosses above the origin.
• If \( b \) is zero, the line passes through the origin itself.
• If \( b \) is negative, the line intersects below the origin.

The y-intercept gives us a starting point for graphing a line and can often represent an initial condition in applied problems. For example, if the line represents a business's revenue over time, the y-intercept might show initial earnings even when no time has passed. In our specific solution, the y-intercept is 2, meaning when \( x \) is zero, \( y \) equals 2.