Problem 23 - 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. $$ A(5,-3) ; \quad \text { slope }-4 $$

Step-by-Step Solution

Answer

The general form is \(4x + y - 17 = 0\).

Step 1: Identify the slope-point form equation

The slope-point form of a linear equation is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line. We know the slope \(m = -4\) and the point \(A(5, -3)\).

Step 2: Substitute the point and slope into the equation

Substitute \(x_1 = 5\), \(y_1 = -3\), and \(m = -4\) into the equation: \( y - (-3) = -4(x - 5) \). Simplify this to get \( y + 3 = -4(x - 5) \).

Step 3: Simplify to find the general form

Expand \(-4(x - 5)\) to get \(-4x + 20\). Then simplify the equation to \( y + 3 = -4x + 20 \). Subtract 3 from both sides to get \( y = -4x + 17 \).

Step 4: Convert to general form

Rearrange the equation \( y = -4x + 17 \) into general form \( Ax + By + C = 0 \). Move \(4x\) to the left side to obtain \( 4x + y - 17 = 0 \).

Key Concepts

Slope-Point Form

The slope-point form is an essential tool in understanding linear equations, especially when you're given a specific point and the slope.
This form of the equation is written as:
\( y - y_1 = m(x - x_1) \).

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

Using slope-point form, we can effortlessly substitute the slope and point directly into the equation to express the specific line.
For example, if a line passes through the point \( (5, -3) \) with a slope of \( -4 \), we substitute these values:

• \(m = -4\)
• \((x_1, y_1) = (5, -3)\)

Thus, the slope-point form of this specific line equation becomes:
\( y + 3 = -4(x - 5) \).
This straightforward method allows you to set up the line equation with minimal computation!

General Form Equation

The general form of a linear equation is crucial for understanding all the standard principles of linear equations.
It is represented as:
\( Ax + By + C = 0 \).

• \( A \), \( B \), and \( C \) are constants.
• \( A \) and \( B \) should not both be zero.

This form is particularly useful because it standardizes the equation, making comparisons between different lines more straightforward.
To convert from slope-point form to general form, you'd typically rearrange the terms.
Consider the equation derived earlier, \( y = -4x + 17 \).
To convert this to the general form, rearrange the terms:

• Move \(-4x\) to the left of the equation, resulting in \( 4x + y - 17 = 0 \).

This newly arranged form is in the standard structure, revealing key information such as slope and axis intersections.

Linear Algebra

In mathematics, linear algebra provides a framework for understanding systems of linear equations and transformations.
It deals extensively with lines, planes, vector spaces, and matrix transformations.
In relation to linear equations, particularly lines in two-dimensional space:

• The slope indicates the steepness and direction of a line.
  
• The point-slope and general forms help in visualizing and plotting these lines efficiently.
  
• Linear algebra principles further apply when examining the interaction of multiple lines, such as in intersection problems.

Being adept in linear algebra can significantly boost one's understanding of linear equations beyond just plotting lines.
It aids in solving systems of equations, determining collinearity, and exploring applications with greater dimensions.
The principles of linear algebra underline not only basic line equation conversions but also the broader applications in fields like computer graphics and physics.