Problem 10 - Introductory and Intermediate Algebra for College Students 4th

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises \(1-28 .\) Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=-1,\) passing through \(\left(-4,-\frac{1}{4}\right)\)

Step-by-Step Solution

Answer

The point-slope form of the line equation is \(y + 1/4 = -x -4\). The slope-intercept form of the line equation is \(y = -x - 4 + 1/4\).

Step 1: Apply the Point-Slope Form

Using the point-slope form of a line equation \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the given point. Substituting the given values, we get: \(y - (-1/4) = -1(x - (-4))\).

Step 2: Simplify the Resulting Equation

Simplify the equation resulted from step 1 to find the point-slope form. The equation can be simplified as: \(y + 1/4 = -x -4\).

Step 3: Convert to Slope-Intercept Form

Rearrange this equation to slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Rearrange the equation obtained from step 2 to finally get: \(y = -x - 4 + 1/4\).

Key Concepts

Slope

The concept of "slope" is fundamental to understanding linear equations. It describes the steepness or incline of a line. In mathematical terms, slope is often represented by the letter \(m\). It represents the ratio of the change in the \(y\)-coordinate (vertical change) to the change in the \(x\)-coordinate (horizontal change) between two distinct points on the line. Mathematically, this is expressed as:\[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\]

• Positive slope: The line inclines upwards as you move from left to right.
• Negative slope: The line inclines downwards as you move from left to right.
• Zero slope: The line is perfectly horizontal, indicating no incline or decline.
• Undefined slope: The line is vertical, indicating an infinite slope.

A handy trick is to remember that "rise over run" helps visualize the slope on a graph. For example, if the slope is \(-1\), this means for every unit you move to the right, you move one unit down.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most intuitive ways to express the equation of a line. It is written as:\[y = mx + b\]In this formula:

• \(y\) is the dependent variable or output.
• \(x\) is the independent variable or input.
• \(m\) is the slope of the line.
• \(b\) is the y-intercept, or where the line intersects the y-axis.

The beauty of the slope-intercept form is its simplicity—it immediately tells you the slope of the line and where it crosses the y-axis.
This format is especially helpful for graphing linear equations because you can easily plot the y-intercept \((0, b)\) and use the slope \(m\) to find other points on the line.

Equation of a Line

The equation of a line connects the relationship between the \(x\) and \(y\) coordinates across all points on the line. There are several forms to express this equation, but the most commonly used is the slope-intercept form \(y = mx + b\). However, another useful form for writing the equation of a line is the point-slope form:\[y - y_1 = m(x - x_1)\]In this equation:

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

This form is incredibly helpful when you have a point on the line and the line's slope. You can start by plugging these values into the equation.
From here, it's easy to rearrange into other forms like slope-intercept, making the point-slope form an excellent bridge between finding and expressing the equation of a line in multiple ways.