Problem 84 - College Algebra

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 equation of a line in slope-intercept form \(y = m*x + b\), can be derived from the point-slope form \(y - y_1 = m(x - x_1)\) by expanding the latter and grouping the terms appropriately to match the structure of the slope-intercept form.

Step 1: Understand the Point-Slope Form of a Line

The point-slope form of a line is given by the equation \(y - y_1 = m(x - x_1)\). Here, \(m\) is the slope of the line and \((x_1, y_1)\) are the coordinates of a specific point on the line.

Step 2: Rearrange the Equation to Get the Slope-Intercept Form

The slope-intercept form of a line is \(y = mx + b\). To transition from the point-slope form to the slope-intercept form, start by expanding the equation into \(y = m*x - m*x_1 + y_1\). Now, the term \(- m*x_1 + y_1\) is equivalent to the y-intercept \(b\), thus simplifying the equation to \(y = mx + b\).

Step 3: Check Your Work

To confirm that the simplification in Step 2 is accurate, substitute \(x_1\) and \(y_1\) in the final equation \(y = mx + b\). This should lead back to the initial point-slope form \(y - y_1 = m(x - x_1)\), thus proving the derivation.

Key Concepts

Point-Slope Form

In the world of linear equations, the point-slope form is a straightforward, useful way to express a line. It is formulated as \(y - y_1 = m(x - x_1)\). This form uses a specific point on the line, represented by \((x_1, y_1)\) and the slope \(m\) to specify the line's unique path.
The core idea here is that it utilizes both a slope and a distinct point to determine the position and direction of the line. By knowing these two pieces of information:

• the slope \(m\) describes how steep the line is, and
• the point \((x_1, y_1)\) tells you exactly where the line passes.

This form is handy because it gives a direct relationship between any point on the line and its slope.

Linear Equations

Linear equations are a cornerstone of algebra and represent relationships where the variables appear in the first degree (that is, they are not raised to any power other than one). A linear equation yields a straight line when graphed on a coordinate plane, highlighting its defining characteristic.
Generally, these equations can be expressed in different forms, such as

• point-slope form: \(y - y_1 = m(x - x_1)\), or
• slope-intercept form: \(y = mx + b\)

These forms are different methods to describe the same linear relationship, and each is useful in its own right depending on the given information and what you need to find.

Slope

In any discussion about lines, the slope is a crucial element. The slope, denoted as \(m\), measures the steepness and direction of a line. Think of it as the line's **tilt**. Would it go uphill, downhill, or remain flat?
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Key things to remember about slope:

• A positive slope means the line ascends as it moves from left to right,
• a negative slope implies it descends,
• zero slope indicates a perfectly horizontal line, and
• an undefined slope means a vertical line.

Understanding slope is essential as it's a common thread in both point-slope and slope-intercept forms of linear equations.

Y-intercept

The y-intercept is where the line crosses the y-axis. This specific point is crucial in determining the line's position in its graph.
In the slope-intercept form expressed as \(y = mx + b\), the \(b\) represents the y-intercept. It shows the value of \(y\) when \(x\) is zero, giving a direct visual marker on the graph.
Useful facts about y-intercepts:

• It helps in sketching the line quickly on the coordinate plane.
• Knowing the y-intercept simplifies graphing when combined with the slope. You start at the y-intercept and use the slope to determine the next points on the line.
• It's a critical part of converting equations from point-slope to slope-intercept form since it transforms the equation's structure into an intuitive graphable form.

The y-intercept provides a starting point, literally and figuratively, for understanding and visualizing linear relationships.