Problem 18 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-2,\) passing through (0,-3)

Step-by-Step Solution

Answer

So, the equation of the line in point-slope form is \( y + 3 = -2x \) and in slope-intercept form is \( y = -2x -3 \).

Step 1: Writing in Point-Slope Form

The form of the point-slope equation is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) are the coordinates of the known point. For this exercise, substituting \(m = -2\) and the point (0,-3) into the point-slope equation gives:\(y - (-3) = -2(x - 0)\), which simplifies to \(y +3 = -2x\).

Step 2: Writing in Slope-Intercept Form

The slope-intercept form of an equation is \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. Using the point-slope form from step 1, we continue to simplify it to the slope-intercept form: \(y = -2x -3 \) by subtracting 3 on both sides.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential for graphing and analyzing linear equations. It is expressed as (Y = mx + b), where (m) is the slope of the line and (b) is the y-intercept, which is the point where the line crosses the y-axis.

In the context of the provided exercise, the slope-intercept form makes it simple to identify these key characteristics of the line. We were given a slope (m = -2) and a y-intercept at (b = -3). Substituting these values into the general slope-intercept equation gives (y = -2x - 3). This tells us that the line falls 2 units in the y-direction for every 1 unit it moves in the x-direction, and it intersects the y-axis at -3.

The slope-intercept form is particularly useful as it allows for quick sketching of the line on a coordinate plane and forms the basis for understanding the behavior of linear relationships. In addition, it's used extensively when dealing with linear modeling, economics, physics, and solving everyday problems involving rates, such as speed, making it a vital concept in many fields.

Linear Equations

Linear equations form the foundation of algebra and are utilized across various scientific disciplines. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in various forms, including (point-slope form), (slope-intercept form), and (standard form).

The fundamental property of linear equations is that they produce a straight line when graphed on a coordinate plane. The constants in the equations dictate the slope and position of this line. For the given exercise, we created a linear equation based on the conditions provided - a slope of (-2) and passing through the point ((0,-3)).

The beauty of linear equations lies in their simplicity and the ease with which they can be manipulated to extract useful information such as slope, y-intercept, and x-intercept, allowing for a deeper understanding of the linear relationship depicted.

Equation of a Line

The equation of a line is a mathematical way of describing the exact linear path that a line takes on a coordinate plane. There are multiple forms to represent these equations, and each form offers different advantages depending on the information that is known.

The point-slope form, used in our initial step of the exercise, is represented as (y - y1 = m(x - x1)). This formula is particularly useful when we have the coordinates of one point on the line ((x1,y1)) and the slope (m). Even when the problem only provides one point and requires finding the equation of the line, point-slope form becomes a direct tool to use.

For the linear equation in our example, knowing just the slope and a single point was sufficient to determine the precise equation that describes the line graphically. Converting this to the more widely known slope-intercept form allows us to quickly understand the behavior of the line with respect to the y-axis. Despite the different forms, each one interconnects, providing coherent methods to describe the geometric and algebraic properties of lines.