Problem 7 - 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 \(=-12,\) passing through \((-8,0)\)

Step-by-Step Solution

Answer

The point-slope form of the equation is \(y = -12(x + 8)\) and the slope-intercept form of the equation is \(y = -12x - 96\).

Step 1: Substituting in point-slope form

Substitute \(m = -12\) and \((x_1, y_1) = (-8, 0)\) in the point-slope form equation. This results in \(y – 0 = -12(x – (-8))\).

Step 2: Simplifying the equation

Simplify this equation to translate into a more usable form. The equation can be simplified to \(y = -12(x + 8)\).

Step 3: Rewrite in slope-intercept form

Rewrite the equation into the slope-intercept form. Distribute -12 to \(x\) and 8, resulting in \(y = -12x - 96\).

Step 4: Final equation form

Both the point-slope form and slope intercept form are now known as: \(y = -12(x + 8)\) and \(y = -12x - 96\), respectively.

Key Concepts

Slope-Intercept Form Equation

The slope-intercept form equation is a way to represent a straight line in coordinate geometry. It is expressed as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis. This form is particularly useful because it provides a quick way to graph the line by using the slope and y-intercept.

In the given exercise, after simplifying the point-slope form, we obtain the slope-intercept form \( y = -12x - 96 \). Here, the slope \( m \) is -12, which tells us that for every unit the x-coordinate increases, the y-coordinate decreases by 12 units. The y-intercept \( b \) is -96, meaning the line crosses the y-axis at the point \( (0, -96) \).

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality between two expressions. They usually involve variables and constants, and solving an algebraic equation means finding the value of the variables that make the statement true. In linear algebraic equations, the highest power of the variable is one.

The initial form we used in the exercise, the point-slope form, is, in essence, an algebraic equation: \( y - y_1 = m(x - x_1) \). By substituting known values of slope (\( m \)) and a point through which the line passes (\( x_1, y_1 \)), we transformed the abstract algebraic equation into a specific linear equation representing a line on a graph.

Linear Equations

Linear equations represent lines in a two-dimensional plane, and their standard forms include: point-slope form, slope-intercept form, and standard form. The defining characteristic of a linear equation is its constant slope, meaning it graphically produces a straight line.

Through the exercise, we started with the slope and a point to derive the equation of the line in point-slope form and converted it to the slope-intercept form. In both cases, the equations are linear because they graph to straight lines. Notably, linear equations are fundamental in various mathematical concepts and are applied widely in disciplines like physics, economics, and engineering.