Problem 6 - Applied Calculus

Question

Determine the slope and the \(y\) -intercept of the line whose equation is given. $$3 x+2 y=8$$

Step-by-Step Solution

Answer

Slope: \(-\frac{3}{2}\); y-intercept: 4.

Step 1: Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. Start with the given equation: \[3x + 2y = 8\]Subtract \(3x\) from both sides to isolate the \(y\)-term on one side:\[2y = -3x + 8\]

Step 2: Solve for y

To further simplify the equation into the slope-intercept form \(y = mx + b\), divide every term by 2:\[y = -\frac{3}{2}x + 4\]

Step 3: Identify the Slope and y-Intercept

Now that the equation is in slope-intercept form, \(y = -\frac{3}{2}x + 4\), identify the slope as \(m = -\frac{3}{2}\) and the y-intercept as \(b = 4\).

Key Concepts

Understanding the Slope of a Line

The slope of a line is a measure of its steepness and direction. It is expressed as the ratio of the rise (vertical change) to the run (horizontal change) between two distinct points on the line. The slope is denoted by the letter **m** in the slope-intercept form of a line: \[ y = mx + b \]Key points to remember about the slope:

• A positive slope means the line is rising from left to right.
• A negative slope means the line is falling from left to right.
• A slope of zero indicates a horizontal line.
• An undefined slope corresponds to a vertical line.

In the given exercise, the term \(-\frac{3}{2}x\) in the slope-intercept form represents the slope \( m = -\frac{3}{2} \). This tells us that for every 2 units the line moves horizontally, it drops by 3 units. The negative sign indicates a downward trend.

Grasping the y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In other words, it's the value of \(y\) when \(x = 0\). In the equation of a line written in slope-intercept form, \(y = mx + b\), the y-intercept is represented by the letter **b**. Here’s what to remember about the y-intercept:

• It provides the starting point of the line on the y-axis when graphing.
• The y-coordinate of this point is given directly by **b**.

For the equation from the exercise, \(y = -\frac{3}{2}x + 4\), the y-intercept is \(b = 4\). This means the line will cross the y-axis at the point (0, 4). It’s an essential feature for plotting the line accurately.

Breaking Down Linear Equations

Linear equations are algebraic equations of the first degree, meaning they only include terms up to the power of one. One of the most common forms of linear equations is the slope-intercept form, \(y = mx + b\), where:

• \(m\) represents the slope (steepness) of the line.
• \(b\) represents the y-intercept (the point where the line crosses the y-axis).

To convert a linear equation into the slope-intercept form, follow these steps:

• Isolate the y-term on one side of the equation.
• Simplify by solving for \(y\) in terms of \(x\).

The exercise originally presented the equation as \(3x + 2y = 8\). By restructuring it into \(y = -\frac{3}{2}x + 4\), we express it in slope-intercept form. This makes it simpler to identify both the slope and the y-intercept, allowing easier graphing and interpretation of the line. Linear equations in this form readily provide insight into the line’s behavior on a graph.