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Chapter 1: Problem 41

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((7,3)\) and perpendicular to the vertical line passing through \((-1,-7)\)

Short Answer

Expert verified
The equation of the line is \(y = 3\).

Step by step solution

01

Identify the Slope of the Vertical Line

A vertical line has an undefined slope. It is of the form \(x = c\), where \(c\) is a constant. The vertical line in question passes through \((-1, -7)\) and has the equation \(x = -1\).
02

Determine the Slope of the Perpendicular Line

The slope of a line perpendicular to a vertical line is horizontal, which means the slope is \(0\). Therefore, the line perpendicular to \(x = -1\) is a horizontal line.
03

Write the Equation of the Horizontal Line

A horizontal line has the equation \(y = b\), where \(b\) is the y-coordinate of any point on the line. Since this line passes through \((7,3)\), the equation is \(y = 3\).
04

Put the Equation in Standard Form

The standard form of a line's equation is \(Ax + By = C\). For the horizontal line \(y = 3\), adjust it to standard form by making \(A = 0\), \(B = 1\), and \(C = 3\). Thus, the equation is currently \(0x + 1y = 3\), or simply \(y = 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form
Understanding the standard form of a line can be really helpful when dealing with different types of equations. The standard form of a line's equation is given by the format \( Ax + By = C \), where:
  • \( A \), \( B \), and \( C \) are integers.
  • \( A \) should not be negative.
  • \( A \) and \( B \) should not both be zero.

This format is often used because it's a universal way to express linear equations. In our example, we had a horizontal line given by \( y = 3 \). To convert it to standard form, we see that \( A = 0 \), \( B = 1 \), and \( C = 3 \).
This conversion might seem trivial for a horizontal line, but it helps when consistent formatting is needed across different equations.
Perpendicular Lines
Perpendicular lines have distinct properties that define their relationship. When two lines are perpendicular, the product of their slopes is \(-1\).
However, there's a special case: when one of the lines is vertical, its slope is undefined. This means:
  • A vertical line with the equation \( x = c \), where the slope is undefined.
  • A line perpendicular to a vertical line is horizontal with a slope of \( 0 \).

In the exercise, the line we were given is perpendicular to the vertical line \( x = -1 \). This tells us the needed line is horizontal, with a slope of zero. This nature of perpendicular lines helps us quickly determine the required direction of our new line.
Horizontal Line
A horizontal line is unique in its own simple way. It is flat and runs parallel to the x-axis. The equation for a horizontal line is \( y = b \), where \( b \) is a constant.
This reflects the y-coordinate of any point the line goes through, meaning its vertical position stays the same as it stretches along the x-axis.
For instance, in our exercise, the line passes through the point \((7,3)\). This tells us the equation of the line is \( y = 3 \), indicating that anywhere you look on this line, the y-coordinate is \( 3 \).
Understanding horizontal lines can greatly simplify the process of graphing them and manipulating their equations, especially when combined with their standard form.

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Most popular questions from this chapter

Both the La Plata river dolphin (Pontoporia blainvillei) and the sperm whale ( Physeter macrocephalus) belong to the suborder Odontoceti (individuals that have teeth). A La Plata river dolphin weighs between 30 and \(50 \mathrm{~kg}\), whereas a sperm whale weighs between 35,000 and \(40,000 \mathrm{~kg} .\) A sperm whale is \(\quad\) order(s) of magnitude heavier than a La Plata river dolphin.

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=2^{-x} ; \text { base } 2 $$

When log \(y\) is graphed as a function of log.x, a straight line results. Graph straight lines, each given by two points, on a log-log plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(2,7),\left(x_{2}, y_{2}\right)=(7,2) $$

Phytoplankton converts carbon dioxide to organic com. pounds during photosynthesis. This process requires sunlight. It has been observed that the rate of photosynthesis is a function of light intensity: The rate of photosynthesis increases approx imately linearly with light intensity at low intensities, saturates at intermediate levels, and decreases slightly at high intensities. Sketch a graph of the rate of photosynthesis as a function of light intensity.

Logistic Transformation Suppose that $$ f(x)=\frac{1}{1+e^{-(b+m x)}} $$ where \(b\) and \(m\) are constants. A function of the form (1.15) is called a logistic function. The logistic function was introduced by the Dutch mathematical biologist Verhulst around 1840 to describe the growth of populations with limited food resources. (a) Show that $$ \ln \frac{f(x)}{1-f(x)}=b+m x $$ This transformation is called the logistic transformation. (b) Given some data, a table of values of \(x\), and the function \(f(x)\) at each value \(x\), explain how to plot the data to produce a straight line, and to estimate the constant \(b\) and \(m\) from the model.
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