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Chapter 2: Problem 26

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

Short Answer

Expert verified
The equation is \(5x - y - 2 = 0\).

Step by step solution

01

Understand the point-slope form

The point-slope form of a line's equation is given by the formula: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. In this task, \(A(0, -2)\) is our point and 5 is the slope. This information will be substituted into the point-slope form.
02

Substitute the given point and slope into the formula

Substitute \((x_1, y_1) = (0, -2)\) and \(m = 5\) into the point-slope form equation: \(y - (-2) = 5(x - 0)\). This simplifies to \(y + 2 = 5x\).
03

Rearrange to the general form of the line equation

To express the equation in general form (\(Ax + By + C = 0\)), rearrange \(y + 2 = 5x\) to get \(5x - y - 2 = 0\). This is the general form of the equation for the line passing through the point \(A(0, -2)\) with slope 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form
The "point-slope form" is a straightforward way to write the equation of a line when you know one point on the line and the line's slope. It's like having a roadmap that lets you convert specific information into a clear equation. The format of the equation is given by:\[ y - y_1 = m(x - x_1) \]Here:
  • \((x_1, y_1)\) is a known point on the line.
  • \(m\) is the slope of the line.
When given point \((0, -2)\) and a slope of 5, you insert these values into the formula:\[ y - (-2) = 5(x - 0) \]This simplifies to:\[ y + 2 = 5x \]Point-slope form is very useful because it provides a direct way to plug in real-world data points and slopes to generate the equation of the line.
General Form of a Line
The "general form of a line" is another way to express the equation of a line. It is written in the format:\[ Ax + By + C = 0 \]This form provides a standardized way of writing linear equations and removes any fractions or decimals that might be present in other forms. To convert from the point-slope form like \(y + 2 = 5x\) to the general form, you'll rearrange terms:
  • Move all terms to one side of the equation to start with zero on the other side.
  • In our example, subtract 5x and 2 from both sides: \[5x - y - 2 = 0\]
The complete and compact nature of the general form makes it convenient for checking line properties such as parallelism or perpendicularity later on.
Slope
The "slope" of a line is a key concept in understanding line equations. It is a measure of how steep the line is. Mathematically, the slope is represented by the letter \(m\) and is calculated as:\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]The slope indicates the direction and angle of the line:
  • A positive slope (like 5 in our example) means the line rises upwards.
  • A negative slope would suggest the line slopes downwards.
  • A slope of zero indicates the line is perfectly horizontal.
In the exercise, knowing the slope is 5 provided the rate at which the line rises as it moves horizontally, setting the stage for writing the line's equation in point-slope form and eventually in general form.

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