Problem 29 - 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. $$ A(2,-4) ; \text { parallel to the line } 5 x-2 y=4 $$

Step-by-Step Solution

Answer

The general form of the line is \( 5x - 2y = 18 \).

Step 1: Understand the given information

We are given a point \( A(2, -4) \) and a line with equation \( 5x - 2y = 4 \). We need to find the equation of a line that passes through the point \( A \) and is parallel to the given line.

Step 2: Recall the slope-intercept form

The slope-intercept form of a line is \( y = mx + b \) where \( m \) is the slope. We need to identify the slope of the given line to determine the slope of the parallel line.

Step 3: Determine the slope of the given line

The equation of the given line is \( 5x - 2y = 4 \). Re-arrange into slope-intercept form: \( -2y = -5x + 4 \), then \( y = \frac{5}{2}x - 2 \). Thus, the slope \( m \) is \( \frac{5}{2} \).

Step 4: Use point-slope form for the parallel line

Since parallel lines have the same slope, set the slope of our new line to \( \frac{5}{2} \). Use the point-slope formula, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point given (2, -4) and \( m = \frac{5}{2} \).

Step 5: Plug values into the point-slope form

Substitute \( x_1 = 2 \), \( y_1 = -4 \), and \( m = \frac{5}{2} \) into the point-slope formula: \( y + 4 = \frac{5}{2}(x - 2) \).

Step 6: Simplify to general form

Expand and simplify: \( y + 4 = \frac{5}{2}x - 5 \). Rearranging gives \( y = \frac{5}{2}x - 9 \). Convert to general form by removing fractions and ensuring all terms are on one side: multiply through by 2 to obtain \( 2y = 5x - 18 \) which rearranges to \( 5x - 2y = 18 \).

Key Concepts

Parallel Lines

Parallel lines are lines in the same plane that never intersect, regardless of how far they extend. This is because they have the same slope. When dealing with linear equations, this property is particularly useful. In mathematics, two lines are considered parallel if their slopes are equal.
For example, if one line has the equation \( y = mx + b \), another line with the equation \( y = mx + c \) will be parallel to it. Notice that both lines share the same slope \( m \), even though their y-intercepts (\( b \) and \( c \)) differ. This means that parallel lines move in the same direction at the same rate. Thus, the concept of parallel lines is pivotal when working with line equations, especially when determining equations that are parallel to a given line.

Slope-Intercept Form

The slope-intercept form of a line’s equation is one of the most intuitive and widely-used formats. It is expressed as \( y = mx + b \). Here, the letter \( m \) represents the slope of the line, while \( b \) is the y-intercept.
Understanding the components:

• The slope \( m \) signifies the steepness or incline of the line. It is calculated as the "rise over run" or the change in \( y \) over the change in \( x \).
• The y-intercept \( b \) is the point where the line crosses the y-axis.

Given a line in the general form, you can convert it to slope-intercept form by solving for \( y \). This way, you can easily identify the line's slope and intercept, thus making it simpler to analyze or compare with other lines.

Point-Slope Formula

The point-slope formula provides a way to write an equation of a line if you know the slope and a point on the line. It is especially handy when deriving the equation of a line parallel or perpendicular to another.
The formula is expressed as: \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a specific point on the line and \( m \) is the slope.

• By substituting the coordinates of the point and the slope, you get an equation that represents the line through that point with the given slope.
• This form is particularly useful when you have a known point and can determine the slope from another source, like a parallel line.

This method streamlines the process of finding a line’s equation, keeping all information directly related to the essential characteristics of that line.

General Form of a Line

The general form of a line's equation is represented as \( Ax + By + C = 0 \), where \( A \), \( B \), and \( C \) are constants. This form standardizes the line's equation for consistency and ease of comparison, which makes it versatile across different problems.
Advantages of the general form:

• All terms on one side set to zero makes it easy to test for the intercepts.
• If \( A \), \( B \), and \( C \) are integers, it is a preferred standard format in certain contexts, like algebraic proofs or calculations.

To convert from the slope-intercept form to the general form, you should rearrange the terms and, if necessary, clear any fractions by multiplying through by a common denominator to keep coefficients as integers. This form serves as a reliable way to present a polynomial equation of a line.