Problem 19 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(--1,\) passing through \(\left(-\frac{1}{2},-2\right)\)

Step-by-Step Solution

Answer

The given line in point-slope form is \(y + 2 = -x - \frac{1}{2}\) and in slope-intercept form is \(y = -x - \frac{5}{2}\).

Step 1: Substitute given values in point-slope form

The provided slope is -1 and the point given is \(-\frac{1}{2}, -2\). Substituting these values into the point-slope form \(y - y1= m(x - x1)\) gives \(y - (-2) = -1(x - (-\frac{1}{2}))\).

Step 2: Simplify the equation

Simplifying the right hand side: \(y + 2 = -1(x + \frac{1}{2})\) and then distributing the -1 on the right hand side gives \(y + 2 = -x - \frac{1}{2}\). This is the equation in point-slope form.

Step 3: Rewrite the equation in slope-intercept form

To write this equation in slope-intercept form \(y = mx + b\), subtract 2 from both sides of the equation from step 2, which gives \(y = -x - \frac{5}{2}\). This is the equation in slope-intercept form.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is foundational in algebra. It is represented as \( y = mx + b \), where \( m \) stands for the slope of the line and \( b \) represents the y-intercept, which is where the line crosses the y-axis.

In the exercise, you're given specific conditions to find both the point-slope and slope-intercept forms. The slope is given as -1, and it passes through the point \(\left(-\frac{1}{2}, -2\right)\). After finding the point-slope form, translating it into the slope-intercept form involves isolating y on one side of the equation. By doing so, we align with the slope-intercept definition. It allows easy graphing of the linear equation and an immediate visualization of the slope and the y-intercept.

Writing Linear Equations

Writing linear equations is a critical skill in algebra that connects various concepts and forms. Point-slope form and slope-intercept form are two methods used to express these equations.

Starting with the point-slope form, \( y - y_1 = m(x - x_1) \), you apply algebraic manipulation to reach the slope-intercept form. To illustrate, you have the slope (m) as -1 and coordinates of a point the line passes through. The goal is to create an equation that represents the line exactly through these parameters. As seen in the solution, you first plug the point and slope into the point-slope form and after some algebraic steps, you end with the more commonly used slope-intercept form. This demonstrates the interconnectedness of different algebraic representations.

Algebraic Equations

Algebraic equations are the core of mathematics. They represent relationships between variables and constants. The exercise given is an ideal example of turning a verbal description into a mathematical model. Here, the constants are the slope and the coordinates of a point.

Algebra involves various forms of the same concept to better suit the context and ease of use. Starting from the point-slope form, which is ideal when a point and slope are known, to the slope-intercept form, which simplifies the visualization of the line on a graph. Each step in solving these equations builds upon the previous one, emphasizing the importance of understanding each algebraic manipulation and its purpose within the larger framework of linear equations.