Problem 24 - 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. $$ A(-1,4) ; \quad \text { slope } \frac{2}{3} $$

Step-by-Step Solution

Answer

The line's equation in general form is \( 2x - 3y + 14 = 0 \).

Step 1: Identify the information given

We are given a point \( A(-1, 4) \) and the slope \( m = \frac{2}{3} \). Our goal is to find the general form of the equation of the line that passes through this point with the given slope.

Step 2: Use the point-slope form of a line

The point-slope form of a line is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \( m \) is the slope. Substituting the given point \((-1, 4)\) and slope \(\frac{2}{3}\), we get:\[ y - 4 = \frac{2}{3}(x + 1) \]

Step 3: Simplify the equation

Expand the equation from Step 2:\[ y - 4 = \frac{2}{3}x + \frac{2}{3} \]Add 4 to both sides to solve for \( y \):\[ y = \frac{2}{3}x + \frac{2}{3} + 4 \]

Step 4: Convert to general form

Add 4 and \(\frac{2}{3}\) to get the constant term:\[ y = \frac{2}{3}x + \frac{14}{3} \]To convert to general form \( Ax + By + C = 0 \), subtract \(y\) from both sides and clear the fractions by multiplying the entire equation by 3:\[ 3y = 2x + 14 \]Rearrange to:\[ -2x + 3y - 14 = 0 \] Thus, the general form of the equation is \( 2x - 3y + 14 = 0 \).

Key Concepts

Point-Slope Form

Point-slope form is a powerful tool for writing the equation of a line when you know one point on the line and its slope. The standard point-slope form is given by the equation \( y - y_1 = m(x - x_1) \). Here, \( (x_1, y_1) \) represents a point on the line, and \( m \) is the slope of the line.

This form is particularly useful because you can immediately plug in the values you know, without needing to rearrange or perform additional calculations. It's like filling in the blanks of a template. In our example, we have a point \( A(-1, 4) \) and a slope \( m=\frac{2}{3} \). Let's see how this works:

• Substitute \( x_1 = -1 \), \( y_1 = 4 \), and \( m = \frac{2}{3} \) into the formula.
• Getting \( y - 4 = \frac{2}{3}(x + 1) \).

This set up allows you to systematically find the equation of a line, making further conversions or calculations easier.

General Form

The general form of a line's equation is another pivotal style, expressed as \( Ax + By + C = 0 \). It offers a neat and standardized way to write line equations, often preferred in situations requiring a unified look or when working with multiple lines. To convert from point-slope form to general form, certain steps are involved.
First, expand the point-slope equation to isolate \( y \):

• From the point-slope form \( y - 4 = \frac{2}{3}(x + 1) \), expand to \( y - 4 = \frac{2}{3}x + \frac{2}{3} \).
• Add 4 to both sides to solve \( y \), resulting in \( y = \frac{2}{3}x + \frac{14}{3} \).

Next, transform it into general form:\

• Subtract \( y \) from both sides and multiply the entire equation by 3 to eliminate fractions: \( 3y = 2x + 14 \).
• Rearrange to fit the general form, obtaining \(-2x + 3y - 14 = 0 \).

Thus, the final general form is \( 2x - 3y + 14 = 0 \), representing the same line, but in a format that’s useful for meeting specific criteria such as clarity and comparison.

Slope Calculation

Slope is often referred to as the 'steepness' or 'incline' of a line and describes how one variable changes in relation to another. It is denoted by \( m \) and can be calculated using the formula \( m = \frac{\Delta y}{\Delta x} \), which translates to \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula indicates how the \( y \) values change as \( x \) changes, using two points \( (x_1, y_1) \) and \( (x_2, y_2) \).
Given only one point and a slope as in our example, you apply this concept directly in the point-slope form rather than recalculate it. The provided slope \( \frac{2}{3} \) indicates that for any 3 units moved horizontally along the line, the vertical change will be 2 units. A positive slope like this shows that the line rises as it moves from left to right.
Understanding slope lets you predict and sketch the direction of a line. It also functions as the axis placeholder in the point-slope formula, providing a seamless way to express line equations.