Problem 8 - Calculus Single Variable

Question

Write the point-slope equation of the line determined by the given data. Slope \(0,\) point \((-\pi, \pi)\)

Step-by-Step Solution

Answer

The point-slope equation of the line is \(y = \pi\).

Step 1: Identifying the Equation Type

We need to use the point-slope form to find the equation of the line. The point-slope form is generally written as \( y - y_1 = m(x - x_1) \), where \( m \) represents the slope and \((x_1, y_1)\) is a point on the line.

Step 2: Substituting the Slope

Given the slope \( m = 0 \), substitute this value into the point-slope formula, which gives \( y - y_1 = 0 \cdot (x - x_1) \). This simplifies to \( y - y_1 = 0 \).

Step 3: Substituting the Point

Insert the point \((-\pi, \pi)\) into the simplified equation. Replace \(y_1\) with \(\pi\), so the equation becomes \( y - \pi = 0 \).

Step 4: Solving for y

To find the equation in slope-intercept form, solve \( y - \pi = 0 \) for \( y \). Adding \(\pi\) to both sides, we get \( y = \pi \).

Key Concepts

Equation of a Line

When we talk about the equation of a line, we typically refer to a mathematical equation that represents a line in two-dimensional space. The equation serves as a clear instruction for locating any point that lies on the line.

There are several forms of line equations, making it adaptable for diverse scenarios. Each form of a line equation is designed to make specific calculations easier based on available data. The most frequently used forms include:

• Point-Slope Form
• Slope-Intercept Form
• Standard Form

In this exercise, we primarily use the point-slope form of the equation to find the line's equation. The point-slope form is perfect when you already have a point on the line and its slope, represented as: \[ y - y_1 = m(x - x_1) \]where \( m \) is the slope and \((x_1, y_1)\) is a specific point on the line. This form allows you to quickly substitute values and solve for a line equation.

Slope-Intercept Form

The slope-intercept form is another common way to write the equation of a line. It provides a simple format to instantly understand crucial properties of the line, such as its slope and its y-intercept. The slope-intercept form is written as follows:\[ y = mx + b \]Here, \( m \) is the slope of the line, and \( b \) is the y-intercept, i.e., the point where the line crosses the y-axis. This form is extremely advantageous for graphing purposes. You can immediately plot the y-intercept on a graph and use the slope to determine the line's angle.

In our step-by-step solution, we converted the point-slope form into the slope-intercept form, resulting in \( y =  \pi \) . This final equation shows that the slope \( m \) is 0, which indicates a horizontal line, and \(b = \pi\), meaning the line crosses the y-axis at \( y =  \pi\).

Linear Equations

Linear equations are fundamental components of algebra and calculus. They describe a constant relationship between variables, which is crucial to understanding more complex mathematical concepts.

Linear equations can model many real-world situations, ranging from calculating expenses to representing physical phenomena. In its purest form, a linear equation results in a straight line when graphed on a coordinate plane.

• A line with a slope of 0, like in our solution, represents a horizontal line. No matter the value of \( x \), \( y \) remains constant.
• Vertical lines cannot be represented by the usual linear equation, as their slope is undefined.

Each part of the linear equation plays an essential role in shaping the line's characteristics. Understanding these components fully will enable you to draw and analyze linear relationships effectively.