Problem 25 - 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. $$ A(4,0) ; \quad \text { slope }-3 $$

Step-by-Step Solution

Answer

The general form of the equation is \(3x + y = 12\).

Step 1: Identify the Point-Slope Formula

A line through a given point \(x_1, y_1\) with a slope \(m\) can be represented by the point-slope form of a linear equation: \[ y - y_1 = m(x - x_1) \] Given that the point is \(A(4, 0)\) and the slope is \(-3\), we recognize this as the problem's starting point.

Step 2: Substitute Values into the Point-Slope Formula

Substitute the given point \(A(4,0)\) and the slope \(-3\) into the point-slope formula: \[ y - 0 = -3(x - 4) \] This simplifies to: \[ y = -3(x - 4) \]

Step 3: Simplify to Find the Equation

Distribute the slope \(-3\) through the expression \(x - 4\): \[ y = -3x + 12 \] This equation can be written in general form as: \\[ y = -3x + 12 \]

Step 4: Rewrite in General Form

To express the equation in the general form \(Ax + By = C\), rearrange the terms: \[ 3x + y = 12 \] So the general form of the equation of the line is \(3x + y = 12\).

Key Concepts

Point-Slope Formula

The point-slope formula is a valuable tool in understanding the relationship between a point and a line's slope. It’s expressed as: \( y - y_1 = m(x - x_1) \), where:

• \( (x_1, y_1) \) is a known point on the line.
• \( m \) is the slope of the line.

This formula allows you to calculate the equation of a line when you know just one point and the slope.
To use the point-slope formula effectively, substitute the specific values of the point and the slope into the formula. For example, with a point \( A(4, 0) \) and a slope of \(-3\), substituted into the formula, it becomes: \( y - 0 = -3(x - 4) \).
The power of the point-slope formula lies in its simplicity. It directly gives the equation of the line without additional calculations, making it easier for both teaching and learning linear equations.

Slope

Slope is a critical concept in linear equations as it measures the steepness or incline of a line. It is often denoted by the letter \( m \) and calculated as the change in \( y \) over the change in \( x \), also known as "rise over run."

• Mathematically, it's expressed as: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
• In the provided exercise, the given slope is \(-3\).
• This negative sign indicates a line that descends from left to right.

The slope tells us how much \( y \) changes for each unit of change in \( x \). If you move one unit to the right along the x-axis, the value of y decreases by 3 units because the slope is -3. Understanding slope is crucial for graphing lines and predicting the direction they move in a coordinate plane.

Equation of a Line

The equation of a line is a way to express the line mathematically. It is usually given in one of several forms, such as point-slope, slope-intercept, or general form. Here’s a quick glance at them:

• Point-Slope Form: \( y - y_1 = m(x - x_1) \)
• Slope-Intercept Form: \( y = mx + b \)
• General Form: \( Ax + By = C \)

In our exercise, we started with the point-slope form: \( y = -3(x - 4) \) derived from the point \( A(4, 0) \) and slope \(-3\). We transformed it into the general form:
Rearranging terms, we have \( 3x + y = 12 \). This is the general form of a linear equation, which is often preferred in mathematics for its simplicity and ability to handle any line on a Cartesian plane.
Understanding each form’s structure helps in moving between different types of equation forms, allowing flexibility in solving and graphing problems involving linear equations.