Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find the slope and y-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify your answers to parts (a) and (b). $$-11-4 y=0$$

Step-by-Step Solution

Answer

The slope of the line is 0 and the y-intercept is -11/4. The line is a horizontal line at \(y = -11/4\) on the graph.

Step 1: Identify the Standard Form of the Line

Rearrange the equation to look like \(y = mx + c\). In this equation, \(m\) corresponds to the slope and \(c\) corresponds to the y-intercept. Rearrange the given equation, which is \(-11 - 4y = 0\), and it becomes \(y = -11/4\). As there's no \(x\) in the equation, \(m\) or the slope is 0 and \(c\), the y-intercept, is -11/4.

Step 2: Draw the Line

To sketch the line, first mark the y-intercept at \(y = -11/4\) on the y-axis. Since the slope of the line is zero, the line will be a horizontal line passing through the y-intercept point, parallel to the x-axis.

Step 3: Verify with Graphing Utility

Enter the equation into a graphing utility, like Desmos or a graphing calculator, to verify the result. The line should appear as a horizontal line at \(y = -11/4\).

Key Concepts

Slope

The slope of a line indicates how steep it is and the direction in which it slants. In linear equations, the slope is often represented by the variable "m" when the equation is in the form of \(y = mx + c\).
In this specific exercise, the equation given is \(-11 - 4y = 0\), which can rearrange to \(y = -11/4\). Here, the 0 means the slope is zero, which tells us that the line is perfectly horizontal.

• Positive Slope: The line goes up as it moves from left to right.
• Negative Slope: The line goes down as it moves from left to right.
• Zero Slope: The line is horizontal, which happens in this exercise.
• Undefined Slope: This occurs in vertical lines.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis. This is represented by the variable "c" in the equation form \(y = mx + c\).
From the simple rearrangement of the equation \(-11 - 4y = 0\) to \(y = -11/4\), we identify the y-intercept as \(-11/4\). So, the line crosses the y-axis at -2.75.

• The y-intercept helps to quickly identify the starting point of a line on a graph.
• It acts as a reference point for sketching the line with the slope.
• In a horizontal line like in this exercise, the y-intercept is also the height of the entire line.

Graphing Utility

A graphing utility is a tool that simplifies drawing and analyzing graphs. Examples include graphing calculators and online tools like Desmos.
These utilities help verify the math work done manually by providing a visual representation. When you input an equation, these tools instantly display the graph and help ensure accuracy.

• Easily plot points and equations with high precision.
• Quickly visualize the nature of the slope and the y-intercept.
• See how changes to equations affect the line's appearance.

Using a graphing utility for the given line equation \(y = -11/4\) confirms that the line is horizontal and correctly shows the y-intercept.

Horizontal Line

A horizontal line is a line that runs left to right across a graph. It has a crucial property where all the y-coordinates are constant but x-coordinates can differ.
In the case of the exercise, where the equation is arranged to \(y = -11/4\), the horizontal line runs parallel to the x-axis, crossing the y-axis at \(-11/4\).

• In a graph, horizontal lines are instantly recognizable because they do not tilt up or down.
• The slope of a horizontal line is always 0, as there is no vertical change when you move along the line.
• Gives a clear indication of constant value with no increase or decrease as x changes.

Horizontal lines can often represent stable or unchanged states in real-world data analysis.