Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-32: Find a general form of an equation of the line through the point \(A\) that satisfies the given condition. $$ A(0,-2) ; \quad \text { slope } 5 $$

Step-by-Step Solution

Answer

The general form of the line is \( 5x - y - 2 = 0 \).

Step 1: Understand the Problem

We are asked to find the general form of the equation of a line. The line passes through the point \( A(0, -2) \) and has a slope of 5.

Step 2: Recall the Point-Slope Form

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

Step 3: Plug Values into Point-Slope Form

Use the given point \((0, -2)\) and the slope \(m = 5\) in the point-slope form. Substituting the values, we get: \( y - (-2) = 5(x - 0) \). Simplify this to \( y + 2 = 5x \).

Step 4: Rearrange to General Form

Our goal is the general form of a line equation, which is \( Ax + By + C = 0 \). From the equation \( y + 2 = 5x \), subtract \(5x\) and \(2\) from both sides to obtain \(-5x + y + 2 = 0 \).

Step 5: Finalize the General Form

Multiply the entire equation by -1 to obtain the more standard general form: \(5x - y - 2 = 0\). Thus, the general form of the line is \(5x - y - 2 = 0\).

Key Concepts

Point-Slope Form

The point-slope form is a way to write the equation of a line when you know the slope and a point on the line. It is brilliantly simple yet powerful. The formula for the point-slope form is \( y - y_1 = m(x - x_1) \). Here, \( (x_1, y_1) \) represents a known point on the line and \( m \) is the slope.

The beauty of point-slope form lies in its intuitive structure. You literally plot a point and use the slope to build the equation around it. This makes it incredibly effective for understanding how lines work in coordinate geometry. For example, if you have a point \((0, -2)\) and a slope of 5, you substitute these values directly into the formula:

• \( x_1 = 0 \), \( y_1 = -2 \).
• \( m = 5 \).

You end up with \( y + 2 = 5(x - 0) \), or \( y + 2 = 5x \) after simplification. This approach lays the groundwork for converting it into other forms, like the general form, quite effortlessly.

General Form

The general form of a line's equation is a more conventional way to express linear equations. It looks like this: \( Ax + By + C = 0 \), where \( A \), \( B \), and \( C \) are constants. This form is handy because it's standardized and works well within systems of equations.

To convert from point-slope to general form, you need to rearrange the equation by gathering all the terms on one side. Take the point-slope form \( y + 2 = 5x \):

• Move \( 5x \) to the left: \(-5x + y + 2 = 0\).
• Multiply through by -1 for a more standard appearance: \( 5x - y - 2 = 0 \).

Achieving the general form involves simple arithmetic manipulations and ensures accuracy across varied calculations and contexts.

Slope

Slope is a fundamental concept that measures how steep a line is. It is represented as \( m \) in equations and is calculated using the ratio of the change in \( y \) (vertical change) to the change in \( x \) (horizontal change). Mathematically, it's expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In practical terms, slope tells you how to move from one point to another along a line. A slope of 5 means that for every one unit you move horizontally (to the right), you move five units vertically (up). This leads to a steep line.

Having the slope allows you to craft the line equation using the point-slope formula. If already known, it simplifies constructing and understanding the line's behavior on a graph. In our example, knowing the slope enabled us to write the equation, showing how slope seamlessly integrates into the broader concept of linear equations.