Problem 60 - College Algebra

Question

a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Graph the equation. $$4 y+28=0$$

Step-by-Step Solution

Answer

The equation, in slope-intercept form, is \(y = -7\). The slope of the line is \(m = 0\) and the y-intercept is \(c = -7\). The line can be graphed as a horizontal line passing through the y-axis at y = -7.

Step 1: Rewrite in slope-intercept form

First, the provided equation \(4y+28=0\) needs to be rearranged into slope-intercept form. This form is represented by \(y = mx + c\). Here, \(x\) is the input variable, \(y\) is the output variable, \(m\) is the slope of the line, and \(c\) is the y-intercept. To get this form, we will subtract 28 from both sides and then divide the equation through by 4. Doing this gives us: \(y = -7\). This is in the slope-intercept form, where \(m = 0\) (since there is no \(x\) term) and \(c = -7\).

Step 2: Provide the slope and y-intercept

From the slope-intercept form of the equation, we can directly read off the slope and the y-intercept. Here, the line has a slope \(m = 0\) because there is no x-term in our equation. The y-intercept \(c = -7\), which means the line crosses the y-axis at the point (0, -7).

Step 3: Graph the equation

A graph can be plotted for this equation. As it is a line with slope 0 and y-intercept -7, it will be a horizontal line passing through the y-axis at y = -7. The line will be parallel to the x-axis at y = -7.

Key Concepts

Equation Rearrangement

Rearranging an equation into slope-intercept form is like fitting it into a specific template. The slope-intercept form is represented as \( y = mx + c \). Here, \( m \) is the slope of the line, and \( c \) is the y-intercept. From this, the form allows us to quickly grasp the line's characteristics and how it behaves on a graph.

To arrange our original equation \( 4y + 28 = 0 \) into the slope-intercept form, we need to solve for \( y \).

• Firstly, isolate the \( y \) term by subtracting 28 from both sides, resulting in \( 4y = -28 \).
• Next, divide every term by 4 to solve for \( y \), giving you \( y = -7 \).

This process gives us the final form of \( y = -7 \), indicating that the line is horizontal and does not depend on \( x \). Thus, our line expressed in slope-intercept form is simple yet quite informative about its nature.

Slope and Y-Intercept

The slope and the y-intercept are key elements that define the behavior of a linear equation. In our example, after transforming the equation to \( y = -7 \):

• **Slope (\( m \))**: This line’s equation does not include an \( x \) term, which means the slope \( m = 0 \). A zero slope indicates a perfectly horizontal line, showing no rise as it moves left or right across the graph.
• **Y-Intercept (\( c \))**: The y-intercept is \( -7 \). This means the line crosses the y-axis at the point \((0, -7)\). It's where the line would "touch" the y-axis, hence it’s a critical point to graph.
  

Understanding these two elements is crucial because they solely define how the line will look on a graph and how it will interact with other potential lines.

Graphing Linear Equations

Graphing a linear equation offers a visual insight into its components and behavior. Let's graph \( y = -7 \):

• This equation is quite straightforward with its zero slope \( m = 0 \), representing a flat line, and having no \( x \) term.
• The y-intercept at \( c = -7 \) dictates that the line will intersect the y-axis at the point (0, -7).

This results in a horizontal line parallel to the x-axis because no matter what \( x \) value is chosen, \( y \) will always be \(-7\). Such visualization can significantly enhance your comprehension of a linear equation's form, especially when it's as simple as a horizontal line. When you see the graph, you'll instantly recognize the consistency this line maintains across all points.