Problem 7 - Calculus Single Variable

Question

Write the point-slope equation of the line determined by the given data. Slope \(-1,\) point \((-\sqrt{2}, 0)\)

Step-by-Step Solution

Answer

The point-slope equation is \( y = -x - \sqrt{2} \).

Step 1: Identify the Formula

The point-slope form equation of a line can be written as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line.

Step 2: Substitute Given Values

We substitute the given values into the point-slope formula. Here, the slope \( m \) is \(-1\), and the point \((x_1, y_1)\) is \((-\sqrt{2}, 0)\). The equation becomes \( y - 0 = -1(x - (-\sqrt{2})) \).

Step 3: Simplify the Equation

Simplify the equation by multiplying and distributing the slope: \( y = -1(x + \sqrt{2}) \). This expands to \( y = -x - \sqrt{2} \), which is the point-slope form of the equation.

Key Concepts

Equation of a Line

A line can be represented by various equations, but one of the most insightful is the point-slope form. This equation of a line is valuable for understanding how a line behaves just by knowing one of its points and the slope. The point-slope form is written as:

• \( y - y_1 = m(x - x_1) \)

In this formula:

• \( m \) represents the slope of the line.
• The point \((x_1, y_1) \) is any specific point on the line.

The point-slope form is particularly useful in algebra because it lets us convert information about the slope and a specific point into a clear equation. This form emphasizes the concept that a line’s position can be easily altered by changing its slope or shifting its points.

Slope

The slope of a line is a key concept in understanding linear equations and their graphs. It measures how steep a line is and is defined as the ratio of vertical change (rise) to horizontal change (run) between two points on the line. The formula to calculate the slope \( m \) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

The slope can tell us a lot about the line:

• A positive slope means the line goes upwards as you move from left to right.
• A negative slope, like in the given exercise example, means the line descends as you go from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope means a vertical line.

In our problem, the slope is

• -1, signifying a line descending at a 45-degree angle relative to the axes.

Understanding the slope is essential because it determines the line's direction and steepness, shaping how it interacts with the coordinate plane.

Substitution Method

The substitution method is a straightforward approach often used to find specific values or to insert known values into equations. In the context of linear equations, it's particularly useful for customizing general equations to fit specific situations.Here's how the substitution method works in our exercise:

• Identify the values given: the slope \( m = -1 \) and the point \( (x_1, y_1) = (-\sqrt{2}, 0) \).
• Substitute these values into the point-slope formula \( y - y_1 = m(x - x_1) \).
• Replace \( y_1 \) with 0, \( m \) with -1, and \( x_1 \) with \(-\sqrt{2} \), modifying the equation accordingly.
• After substitution, simplify the equation as necessary. For our example, this results in \( y = -x - \sqrt{2} \).

This method is a powerful algebraic technique that transforms an abstract equation into something that fits your specific data. With the substitution method, you can easily take theoretical models and apply them to real-world scenarios.