Problem 24 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Passing through \((-2,-5)\) and \((6,-5)\)

Step-by-Step Solution

Answer

The point-slope form and slope-intercept form of the line passing through \((-2,-5)\) and \((6,-5)\) is \(y=-5\).

Step 1: Calculation of slope

First, calculate the slope of the line using the formula for finding the slope between two points which is \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). Based on this formula, and using the given points \((-2, -5)\) and \((6, -5)\), the slope \(m\) can be calculated as \(m=\frac{-5-(-5)}{6-(-2)}=0\), as the numerator is zero.

Step 2: Write point-slope form

The point-slope form of the equation is given by \(y-y_{1}=m(x-x_{1})\). Substituting for \(m\), \(x_{1}\), and \(y_{1}\) with 0, -2, and -5 respectively, we get \(y-(-5)=0(x-(-2))\). This simplifies to \(y=-5\).

Step 3: Write slope-intercept form

The slope-intercept form of an equation is given by \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, as the slope (\(m\)) is 0 and \(b\) (y-intercept) is -5, the slope-intercept form of the equation becomes \(y=0x-5\) which simplifies further to \(y=-5\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common and useful representations in algebra. It's represented as \(y = mx + b\). This formula allows you to quickly identify two important characteristics of a line:

• The slope \(m\), which tells you the steepness of the line.
• The y-intercept \(b\), which is the point where the line crosses the y-axis.

Usually, this form is favored in algebra because it gives a direct way to graph a line and understand its behavior. If \(m = 0\), the line is horizontal, indicating there's no vertical change as you move along the x-axis. This means the equation is simply \(y = b\), a constant line that runs parallel to the x-axis. The example given in the original exercise involves such a line, as it demonstrates a line with a slope of 0, resulting in an equation \(y = -5\). This highlights how useful the slope-intercept form can be in quickly identifying the nature of a line.

Slope Calculation

Slope calculation is a crucial step in understanding the behavior of a line. The slope, often denoted by \(m\), is a measure of how steep a line is and can be calculated between two points \((x_1, y_1)\) and \((x_2, y_2)\) using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula looks at the "rise" (vertical change) over the "run" (horizontal change) between two points. In the case of the points \((-2, -5)\) and \((6, -5)\), the slope calculated is \(m = \frac{-5 - (-5)}{6 - (-2)} = 0\).
A slope of 0 indicates that the line is perfectly horizontal. This tells us that no matter how far you move along the line, the value of \(y\) remains constant. In practical terms, when you see a slope of 0, you'll know it's a straight line without any incline or decline. Understanding this concept helps in easily translating between different forms of linear equations.

Linear Equations

Linear equations form the backbone of algebra and represent relationships that create a straight line when graphed. These equations can be expressed in various forms, such as point-slope, slope-intercept, and standard form. Each of these has its own benefits, but they all describe the same type of relationship.

• Point-Slope Form: \(y - y_1 = m(x - x_1)\) is useful for forming the equation of a line quickly when you know a point and the slope.
• Slope-Intercept Form: \(y = mx + b\) is ideal for immediately identifying the slope and y-intercept, simplifying graphing and comparison of lines.

In the original exercise, both point-slope form and slope-intercept form were used to express the same line equation, \(y = -5\). This equation suggests that every point on the line has the same y-value, indicative of a horizontal line. Recognizing these different forms and understanding their use cases is essential for solving and graphing linear equations with confidence.