Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-34, find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) \text { Through }(3,4) \text { with slope }-1

Step-by-Step Solution

Answer

The equation of the line is \(x + y - 7 = 0\).

Step 1: Understand the Problem

We are tasked with writing the equation of a line that passes through the point (3, 4) with a slope of -1.

Step 2: Apply the Point-Slope Formula

Use the point-slope form of a line equation: \[ y - y_1 = m(x - x_1) \]where \( m \) is the slope and \((x_1, y_1)\) is the point. For this problem, substitute \( m = -1 \), \( x_1 = 3 \), and \( y_1 = 4 \):\[ y - 4 = -1(x - 3) \].

Step 3: Simplify the Equation

Distribute the slope \(-1\) through the equation:\[ y - 4 = -x + 3 \].

Step 4: Rearrange to Standard Form

Rearrange the equation to the standard form \(Ax + By + C = 0\):1. Add \(x\) to both sides: \[ x + y - 4 = 3 \]2. Subtract 3 from both sides to get it to the standard equation:\[ x + y - 7 = 0 \].

Key Concepts

Point-Slope Formula

The point-slope formula is a crucial tool for understanding how to write the equation of a line using a known point on the line and its slope. The formula is given by: \[ y - y_1 = m(x - x_1) \]Here's what you need to know:

• \(y - y_1\) is the difference in the y-values, representing how much the line rises or falls.
• \(m\) is the slope, indicating the steepness of the line, where \(m = \frac{\text{rise}}{\text{run}}\).
• \(x - x_1\) shows the difference in the x-values.

In the given exercise, this formula helps us express the relationship between x and y for the line passing through a specific point \((x_1, y_1) = (3, 4)\) with a slope \(m = -1\). By plugging these values into the formula, we can craft the equation before transforming it into other forms as necessary. This approach makes it straightforward to find a specified point's coordinates in an equation, illustrating the uniformity of linear relationships.

Slope

The slope of a line is a fundamental concept in algebra and geometry that describes the direction and steepness of a line. The slope is usually represented by the letter \(m\) and is defined as:\[ m = \frac{\Delta y}{\Delta x} \] In simpler terms:

• \(\Delta y\) refers to the change in the y-direction (vertical).
• \(\Delta x\) indicates the change in the x-direction (horizontal).

For this problem, the slope is given as \(-1\). This tells us two key things:

• The line decreases, meaning it slants downward from left to right.
• The line will decrease by 1 unit in the y-direction for every 1 unit increase in the x-direction.

Understanding slope helps in visualizing the line, innovatively connecting the rate of change concept as seen in real-world scenarios and high school mathematics. Knowing how to calculate and interpret slope is integral to graphing lines and comprehending linear equations.

Standard Form of a Line

The standard form of a line is presented as:\[ Ax + By + C = 0 \]Here:

• \(A\), \(B\), and \(C\) are integers.
• \(A\) is usually a positive integer.
• \(x\) and \(y\) are the variables representing coordinates on a graph.

The standard form is particularly handy because it communicates clear information about the line's relationship to both axes. It also makes calculations related to intercepts and parallel lines a bit more straightforward. Given the initial point-slope equation \(y - 4 = -x + 3\), converting it to the standard form aids in various mathematical applications. By rearranging and combining like terms, we arrive at \(x + y - 7 = 0\). This transformation aligns with high school mathematics curriculum and presents the equation in a manner ready for broader application and analysis.

High School Mathematics

High School Mathematics encompasses a wide range of fundamental concepts, among which understanding linear equations is essential. As students navigate through this stage, they are introduced to tools for:

• Graphing equations, seeing the behavioral trends, and drawing connections across different mathematical areas.
• Simplifying complex mathematical relationships into understandable components using forms like slope-intercept and point-slope.
• Transforming equations from one form to another, such as converting from point-slope to standard form.

Exercises similar to the one presented, involving points and slopes to derive equations, are pivotal in building these algebraic foundations. These practices help students develop critical analytical skills beneficial for advanced studies in mathematics, science, technology, and beyond. In high school, mastering these concepts prepares learners for future educational endeavors and real-world problem-solving scenarios.