Problem 95 - College Algebra Essentials

Question

Explain how to derive the slope-intercept form of a line's equation, \(y=m x+b,\) from the point-slope form $$y-y_{1}=m\left(x-x_{1}\right)$$

Step-by-Step Solution

Answer

The slope-intercept form \(y=mx+b\) is derived from the point-slope form \(y-y_{1}=m(x-x_{1})\) by distributing the slope to the terms in parentheses and rearranging the equation to isolate \(y\). Therefore, \(b\), the \(y\)-intercept, is equal to \(-mx_{1} + y_{1}\).

Step 1: Understand the point-slope form

The point-slope form of the equation \(y-y_{1}=m(x-x_{1})\) relates to a line passing through the point \((x_{1},y_{1})\) with a slope of \(m\).

Step 2: Transform the point-slope form to slope-intercept form

To derive the slope-intercept form \(y = mx + b\), we rearrange and simplify the point-slope form equation. Start by distributing the slope \(m\) to terms in the parentheses to get \(y-y_{1}=mx-mx_{1}\). Next, bring \(y_{1}\) to the other side of the equation by adding \(y_{1}\) on both sides: \(y = mx - mx_{1} + y_{1}\).

Step 3: Understand the slope-intercept form

The equation now looks like the slope-intercept form \(y=mx+b\), where the slope \(m\) is as it is, and the \(y\)-intercept \(b\) is equal to \(-mx_{1} + y_{1}\). This means that the line crosses the \(y\)-axis at this point \(b\).

Key Concepts

Point-Slope Form

Point-slope form is a way to write the equation of a line using the slope and a specific point on the line. It's written as: \[ y - y_1 = m(x - x_1) \] Where

• \((x_1, y_1)\) is a point on the line
• \(m\) is the slope of the line

This form is particularly useful when you have a known point and the slope. The slope \(m\) represents the steepness or tilt of the line, and the point \((x_1, y_1)\) is where the line passes through in the coordinate plane. By using these values, you can easily plot the line or convert it into other forms, like the slope-intercept form.

Equation of a Line

An equation of a line is a mathematical way of expressing a line on a graph. It allows us to understand how the line behaves by using algebra. Common forms of linear equations include: point-slope form, slope-intercept form, and standard form. The equation describes all points \((x, y)\) that lie on the line.In essence, an equation of a line tells you everything you need to know about the line's orientation (slope) and its location (where it crosses the axes). For practical purposes, once you have the equation, you can plot the line, find intersections with other lines, or solve real-world problems involving linear relationships.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to make them more useful or solve them. Let's see how we manipulate the point-slope form to the slope-intercept form.Start with \[ y - y_1 = m(x - x_1) \]1. Distribute \(m\) across \((x-x_1)\): \[ y - y_1 = mx - mx_1 \]2. Isolate \(y\) by adding \(y_1\) to both sides: \[ y = mx - mx_1 + y_1 \]Here, we've rewritten the equation to show the slope \(m\) and to derive the \(y\)-intercept as \(-mx_1 + y_1\). This method of algebraic manipulation is crucial for converting between different forms of linear equations.

Linear Equations

Linear equations represent straight lines in algebra. These are equations of the first-degree, which means the highest exponent of a variable is one.They can be written in various forms including:

• Standard form: \(Ax + By = C\)
• Slope-intercept form: \(y = mx + b\)

The slope-intercept form is particularly popular because it makes it easy to quickly understand the characteristics of the line. The slope \(m\) shows the rate at which \(y\) changes per unit \(x\), and \(b\), the intercept, illustrates where the line crosses the \(y\)-axis. Recognizing linear equations and understanding their forms allow us for speedy graphing and solving of various algebraic problems.