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

Step-by-Step Solution

Answer

The slope-intercept form of the equation for the given conditions is \(y = -8x - 26\).

Step 1: Apply the point and slope to the formula

Substitute the given values into the point-slope form equation. The slope \(m = -8\) and the point is \((-3, -2)\). So, \(y - (-2) = -8(x -(-3))\). Simplifying this gives us \(y + 2 = -8(x + 3)\).

Step 2: Distribute multiplication

Now, distribute the multiplication to obtain \(y + 2 = -8x - 24\).

Step 3: Transform to slope-intercept form

Lastly, convert to the slope-intercept form of line equation by isolating \(y\). By subtracting 2 from both sides, we obtain \(y = -8x -26\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most popular ways to express the equation of a line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, which is the point where the line crosses the y-axis. This form is incredibly handy because it allows you to quickly identify both the slope and the y-intercept of the line just by looking at the equation.How to Identify Key Features:

• Slope \( (m) \): This tells you how steep the line is. A positive slope means the line rises as it moves to the right, while a negative slope means it falls.
• Y-Intercept \( (b) \): This is the value of \( y \) when \( x = 0 \). It tells you where the line crosses the y-axis.

This form is not only helpful in graphing but also when writing equations based on given data points, like moving from point-slope form to slope-intercept form.

Linear Equations

Linear equations are algebraic expressions of lines in a two-dimensional space and they show a direct relationship between two variables, traditionally \( x \) and \( y \). These relationships are expressed with equations of the highest degree of one, such as \( y = mx + b \). They always form straight lines when graphed on a coordinate plane.Key Characteristics:

• Simplicity: They can often be solved for one variable in terms of the other and are easy to graph.
• Forms: While the slope-intercept form \( y = mx + b \) is common, other forms include standard form \( Ax + By = C \), and point-slope form \( y - y_1 = m(x - x_1) \).
• Applications: These equations are used for real-world phenomena like predictive modeling, economics, and science.

Each linear equation can be transformed between forms for various uses and simplifications, making them versatile tools in algebra.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebra to study geometric problems. It allows us to examine figures in a coordinate plane using a pair of numerical coordinates, which provide a bridge between algebraic equations and geometric shapes.Basic Elements:

• Coordinates: Each point on the plane is represented by an ordered pair \((x, y)\).
• Lines: Defined by linear equations, lines can be analyzed for specific properties such as requiring determination of slopes, midpoints, and lengths.
• Transformation: Shifting between different forms of linear equations to suit various problems is a key skill.

A solid understanding of coordinate geometry provides the groundwork for higher mathematics, such as calculus and helps in solving real-world problems by using of graphical representation.