Problem 17 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(13-26,\) begin by solving the linear equation for \(y .\) This will put the equation in slope-intercept form. Then find the slope and the \(y\) -intercept of the line with this equation. $$6 x+y=0$$

Step-by-Step Solution

Answer

The slope of the given line equation is -6 and the y-intercept is 0.

Step 1: Re-arrange the equation in slope-intercept form

The given equation is \(6x + y = 0\). The first step is to isolate \(y\) to be able to find the slope and y-intercept. This can be done by subtracting \(6x\) from both sides, giving us the equation in slope-intercept form: \(y = -6x + 0\)

Step 2: Identify the slope of the line

In the equation \(y = -6x + 0\), the coefficient of \(x\) (which is -6) is the slope of the line. So, the slope \(m\) is -6.

Step 3: Identify the y-intercept of the line

In the slope-intercept form, the constant that remains after \(y\) has been isolated (that is, 0 in this case) is the y-intercept. So, the y-intercept \(c\) is 0.

Key Concepts

Linear Equations

At the heart of algebra lies the concept of linear equations, which are used to describe relationships where there is a constant rate of change. When dealing with linear equations, our goal is to find the value of the unknown variable that makes the equation true. One common form of a linear equation is the slope-intercept form, which is expressed as
\[ y = mx + b \]
In this equation, \( m \) represents the slope, and \( b \) represents the y-intercept. By rearranging any linear equation into this form, it becomes straightforward to analyze the relationship it represents. This is exactly what the step-by-step solution is illustrating when converting the equation \( 6x + y = 0 \) into slope-intercept form.

Slope of a Line

Understanding the 'slope' of a line is fundamental to grasping how a line moves across the coordinate plane. Mathematically, slope indicates the rate at which the \( y \) value of a line changes relative to the \( x \) value. If you think of it in terms of climbing a hill, the slope tells you how steep the hill is. The mathematical definition of slope is the 'rise over run', which refers to how much the line goes up or down for a horizontal movement to the right.
In the context of the original equation, \( y = -6x \), the slope \( m \) is \( -6 \), indicating that for every unit we move right along the x-axis, the line drops 6 units down. It demonstrates a steep decline, or in other words, the line is descending rapidly.

Y-Intercept

The y-intercept is where the line crosses the y-axis on a graph. It's a specific point, with coordinates \( (0, b) \), where \( b \) is the y-intercept from the slope-intercept form of an equation. The y-intercept provides a starting point for plotting a line on a graph since it is the value of \( y \) when \( x \) is zero.
For the given exercise, after isolating the \( y \) variable, we see that the y-intercept is \( 0 \), represented by the equation \( y = -6x + 0 \). Whenever the y-intercept is zero, the line passes through the origin, which means it starts right at the center of the coordinate grid. Understanding where a line intercepts the y-axis is crucial as it helps to visually represent the linear equation on a graph. It serves as a reference point from which to use the slope to continue plotting the line.