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Chapter 1: Problem 77

In Exercises 65-78, find the slope-intercept form of the equation of the line passing through the points. Sketch the line. \((\frac{7}{3}, -8)\), \((\frac{7}{3}, 1)\)

Short Answer

Expert verified
The slope-intercept form of the line passing through the points (\(\frac{7}{3}, -8)\) and (\(\frac{7}{3}, 1)\) is \( x = \frac{7}{3} \)

Step by step solution

01

Find the Slope

The formula to find the slope (m) when two points \((x_1, y_1)\) and \((x_2, y_2)\) are given is \[ m=\frac{y_2-y_1}{x_2-x_1} \] For the given points \((\frac{7}{3}, -8)\) and \((\frac{7}{3}, 1)\) we can see that the x-values are the same, so it would lead to a division by zero. Therefore, this is the special case of a vertical line and the slope is undefined.
02

Find the y-intercept

Because we're dealing with a vertical line, we don't need to find a y-intercept. Vertical lines don't intersect the y-axis (unless they are the y-axis). They are parallel to it. The equation for a vertical line is in the form \(x = k\), where k is the common x-value of all points on the line, which in our case is \(\frac{7}{3}\).
03

Write the Equation in Slope-Intercept Form

For a vertical line where the slope is undefined and instead, a specific value for x is given. Therefore, the equation in slope-intercept form is \(x = \frac{7}{3}\)

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Key Concepts

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

Equation of a Line
Understanding the equation of a line is crucial for connecting coordinates in any linear fashion. The equation is commonly written in the slope-intercept form, which is \[y = mx + b\]where:
  • \(y\) is the dependent variable (usually represents the vertical position).
  • \(x\) is the independent variable (usually represents the horizontal position).
  • \(m\) is the slope of the line (indicating its steepness or tilt).
  • \(b\) is the y-intercept (the point where the line crosses the y-axis).
This form makes it easy to immediately recognize the slope and the y-intercept of the line. However, not all lines can be represented by this form, especially vertical lines, where the slope is undefined.
Vertical Line
A vertical line is a line that moves straight up and down on the graph. It is parallel to the y-axis and has a distinctive feature: all points on a vertical line have the same x-coordinate. The equation for a vertical line is different from the typical slope-intercept form because it does not have a y-intercept. Instead, it's given by the form:\[x = k\]where \(k\) is the constant x-value for all points on this line. Vertical lines don't cross the y-axis at any particular point unless the entire line is the y-axis itself. In the case where vertical lines are involved, the idea of slope (as a concept of steepness) does not apply in the same way.
Slope
The slope of a line explains the line's direction and steepness. It is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Given two points, the formula for the slope \(m\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]A slope can reveal a lot about a line:
  • A positive slope indicates the line rises from left to right.
  • A negative slope indicates the line falls from left to right.
  • A zero slope indicates a horizontal line.
  • An undefined slope means the line is vertical.
In cases such as vertical lines, the x-values of the points are the same, leading to a division by zero. This is why vertical lines have an undefined slope.
Undefined Slope
An undefined slope occurs when a line is vertical. This concept breaks from the traditional understanding of slope because vertical lines do not "run" left or right. When calculating the slope using two points with the same x-values, the formula\[m = \frac{y_2 - y_1}{x_2 - x_1}\]results in a denominator of zero, which is mathematically meaningless. Because vertical lines do not tilt in ordinary directions (upward or downward in relation to the x-axis), the slope is said to be undefined. This is essential to remember when identifying or graphing vertical lines, as it helps differentiate them from other types that are more easily represented in slope-intercept form.

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Most popular questions from this chapter

In Exercises 49-58, find a mathematical model for the verbal statement. \(y\) varies inversely as the square of \(x\).

In Exercises 109-112, determine if the situation could be represented by a one-to-one function. If so, write a statement that describes the inverse function. The height \(h\) in inches of a human born in the year 2000 in terms of his or her age \(n\) in years.

SALES The total sales (in billions of dollars) for Coca-Cola Enterprises from 2000 through 2007 are listed below. (Source: Coca-Cola Enterprises, Inc.) 2000 14.750 2001 15.700 2002 16.899 2003 17.330 2004 18.185 2005 18.706 2006 19.804 2007 20.936 (a) Sketch a scatter plot of the data. Let \(y\) represent the total revenue (in billions of dollars) and let \(t = 0\) represent 2000. (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the \(regression\) feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the sales of Coca-Cola Enterprises in 2008. (f) Use your school's library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e).

In Exercises 67-74, find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) \(F\) is jointly proportional to \(r\) and the third power of \(s\). (\(F = 4158\) when \(r = 11\) and \(s = 3\).)

THINK ABOUT IT The function given by \(f(x) = k(2 - x - x^3)\) has an inverse function, and \(f^{-1}(3) = -2\). Find \(k\).
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