Problem 5 - 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 \(=-8,\) passing through \((-3,-2)\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -8x - 26\).

Step 1: Identify Given Information

For the problem at hand, the slope of the line \(m\) is given as \(-8,\) and the point that the line passes through (denoted as \((x_1, y_1)\)) is \((-3, -2)\).

Step 2: Insert Values Into Point-Slope Form

The general form of the point-slope equation is \(y - y_1 = m(x - x_1)\). Substituting the given values into this equation, we get \(y - (-2) = -8 (x -(-3)).\

Step 3: Simplify the Equation

Simplifying the equation yields \(y + 2 = -8x -24\). We can isolate \(y\) by subtracting \(2\) from both sides to get \(y = -8x - 26\).

Step 4: Write in Slope-Intercept Form

The derived equation \(y = -8x - 26\) is now in the slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing a linear equation. It is one of the most intuitive forms because it clearly shows two important characteristics of the line: the slope and the y-intercept.
The general formula for the slope-intercept form is:

• \(y = mx + b\)

In this formula, \(m\) represents the slope of the line, while \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
Using the example from the exercise, after converting from point-slope form, the equation becomes \(y = -8x - 26\). Here, the slope \(m\) is \(-8\), indicating a steep decline in the graph as you move along the x-axis.
The y-intercept \(b\) is \(-26\), meaning that the line crosses the y-axis at the point (0, -26). This can help visualize the graph of the line quickly as you know where it starts on the y-axis and how it behaves across the graph.

Linear Equations

Linear equations form the backbone of algebra and are fundamental for high school mathematics. They're simple equations involving two variables, usually \(x\) and \(y\), with no exponents or squared terms.
These equations represent straight lines when plotted on a coordinate grid, hence the name 'linear.' The relationship between the variables is constant; in other words, the change in one variable results in a proportional change in the other.

• The standard form of a linear equation is: \(Ax + By = C\)

This contrasts with the slope-intercept and point-slope forms, offering flexibility depending on the information at hand.
Linear equations are excellent for representing constant rate problems, comparing rates, and they provide a strong foundational understanding for calculus and advanced math concepts.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, provides the tools and language to describe geometric figures using coordinates. It's where algebra meets geometry, allowing us to explore the properties of lines, curves, and shapes.
In coordinate geometry, points are defined in a plane using ordered pairs \((x, y)\). Lines, meanwhile, are described by equations, such as the ones explored in this exercise.
For any two points on a line, the slope \(m\) quantifies their steepness. This feature relates directly to the rise over run principle: \(m = \frac{{y_2-y_1}}{{x_2-x_1}}\). This concept is vital when moving from one concept, like point-slope form, to another, like slope-intercept form.

• The coordinate plane consists of two perpendicular axes, the x-axis and the y-axis.
• It enables precise calculations and detailed geometric descriptions.

Coordinate geometry is an essential part of mathematics education, serving as the foundation for understanding graphs, solving geometry problems with algebra, and much more.