Problem 39 - Precalculus

Question

(a) Find an equation of the line that passes through the origin and is perpendicular to the line \(3 x+4 y=12\) (b) Use a graphing utility to check that your answer in part (a) is reasonable. (That is, graph the two lines using true proportions; the line you found should appear to pass through the origin and be perpendicular to \(3 x+4 y=12 .)\)

Step-by-Step Solution

Answer

The equation is \(y = \frac{4}{3}x\). It's perpendicular to the given line.

Step 1: Identify the slope of the given line

The equation of the line is given in standard form: \(3x + 4y = 12\). First, convert it to slope-intercept form \(y = mx + b\) to identify the slope.\[4y = -3x + 12 \] \[y = -\frac{3}{4}x + 3\] Thus, the slope \(m\) of the line is \(-\frac{3}{4}\).

Step 2: Determine the perpendicular slope

Two lines are perpendicular if the product of their slopes is \(-1\). This means that if one line has slope \(m\), the perpendicular line will have slope \(-\frac{1}{m}\). For the line \(3x + 4y = 12\) with slope \(-\frac{3}{4}\), the perpendicular slope is \[m_{perpendicular} = \frac{4}{3}\]

Step 3: Write the equation of the perpendicular line

A line that is perpendicular to \(3x + 4y = 12\) and passes through the origin \((0, 0)\) will have the form \(y = mx\) where \(m\) is the slope. So, the equation is \[y = \frac{4}{3}x\]

Step 4: Verify using a graphing utility

Using a graphing utility, plot the two lines: \(3x + 4y = 12\) and \(y = \frac{4}{3}x\). Check that the line \(y = \frac{4}{3}x\) passes through the origin and intersects the first line at a right angle. This visual confirmation assures that the solution is correct.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is an essential concept in algebra. It is expressed as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept. This form makes it easy to understand and graph lines because directly from the equation, you can determine:

• The slope: which tells you how steep the line is.
• The y-intercept: which tells you where the line crosses the y-axis.

To convert a line from its standard form (for example, \(Ax + By = C\)) to the slope-intercept form, you solve for \(y\). Suppose we have the line equation \(3x + 4y = 12\), to write it in the slope-intercept form, you rearrange the terms to get \(y = -\frac{3}{4}x + 3\). Here, the slope \(-\frac{3}{4}\) tells us the line decreases by 3 units vertically for every 4 units it moves horizontally. The y-intercept 3 tells us the point where the line intersects the y-axis.
Such transformations are crucial when analyzing and comparing lines, as they allow us to immediately interpret and graph the overall trend of the set of points.

Equation of a Line

An equation of a line provides a mathematical way to describe all the points along that line. When we talk about the line equation considering perpendicularity and specific points, it gets even more interesting. Lines are perpendicular if the product of their slopes is

• -1: This means if you have a line with a slope of \(-\frac{3}{4}\), as in our original example line, its perpendicular counterpart will have a slope \(m_{perpendicular} = \frac{4}{3}\).

This stems from the geometrical principle that perpendicular lines make right angles where they intersect. Knowing this, if a line passes through a specific point, like the origin (0,0), it simplifies the equation to \(y = mx\) because when \(x=0\), \(y\) should also be 0. In essence, understanding the equation of a line in diverse scenarios helps in interpreting graphs, navigating coordinate geometry, and analyzing linear relationships.

Graphing Utility

A graphing utility is a technological aid that helps visualize mathematical concepts and verify calculations. For example, if you're tasked with checking the perpendicularity of two lines, graphing utilities make this task straightforward and intuitive. They allow you to:

• Plot equations: letting you see lines and their intersections on a coordinate plane.
• Check conditions: such as whether the perpendicularity condition between two lines is met by confirming their intersection is at a right angle.

When verifying if the line \(y = \frac{4}{3}x\) is indeed perpendicular to \(3x + 4y = 12\), you can use a graphing utility to plot both lines. The visualization confirms that the calculated slopes are accurate also that the line passes through the origin. These visual confirmations serve as a powerful way to reinforce algebraic solutions and improve understanding for students who benefit from seeing the theoretical concepts come to life.