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Chapter 1: Problem 88

True or False: A vertical line can be expressed in slope-intercept form.

Short Answer

Expert verified
False, a vertical line cannot be expressed in slope-intercept form.

Step by step solution

01

Understand Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.
02

Analyze the Characteristics of a Vertical Line

A vertical line is defined by an equation in the form \( x = c \), where \( c \) is a constant. This indicates that for all values of \( y \), \( x \) remains the same.
03

Compare Vertical Line Equation to Slope-Intercept Form

In slope-intercept form, \( y = mx + b \), the presence of \( x \) in the equation implies a defined slope \( m \). A vertical line does not have a slope (it is undefined) because it does not change in the \( y \) direction for changes in \( x \), hence it cannot be expressed as \( y = mx + b \).
04

Conclusion

Since a vertical line does not fit the \( y = mx + b \) format due to its undefined slope, a vertical line cannot be expressed in slope-intercept form.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a common way to write the equation of a line. It's expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) denotes the y-intercept of the line. This form is particularly useful because it provides immediate insight into how the line behaves on a graph:
  • The slope \( m \) shows the rate at which the line rises or falls as you move along the x-axis. A positive slope indicates an upward trend, while a negative slope indicates a downward trend.
  • The y-intercept \( b \) is the point where the line crosses the y-axis. It gives a clear starting point for graphing when \( x = 0 \).
In essence, the slope-intercept form allows us to quickly sketch and understand the movement of linear lines, making it a powerful tool in algebra and beyond.
Undefined Slope
Understanding slope involves knowing the rate of change between two points on a line, calculated as the rise over run, or change in \( y \) over the change in \( x \). In the case of vertical lines, however, things are a bit different. A vertical line is special because all points on the line have the same x-coordinate, meaning the change in \( x \) is zero. The formula for slope \( m \) is \( m = \frac{\text{rise}}{\text{run}} \). For vertical lines, the run is zero, leading to a situation of dividing by zero, which is undefined in mathematics.
  • Vertical lines are represented by equations of the form \( x = c \), where \( c \) is the constant value of \( x \) for which the line is vertical.
  • No matter how \( y \) changes, \( x \) stays the same, which is why the slope cannot be calculated as a usual number.
Since vertical lines defy the fixed ratio of rise over run, we label their slope as undefined.
Linear Equations
Linear equations are mathematical expressions representing straight lines on a graph. They come in several forms, but the most well-known is the slope-intercept form \( y = mx + b \). However, not all lines fit this form neatly:
  • Slope-Intercept Form: Ideal for lines with well-defined slopes that aren't perfectly vertical or horizontal.
  • Point-Slope Form: Useful when we know one point on the line and the slope, written as \( y - y_1 = m(x - x_1) \).
  • Standard Form: Another variation is \( Ax + By = C \), useful for certain algebraic manipulations.
Vertical lines, such as \( x = c \), technically classify as linear equations though they lack a traditional slope. They stand as a reminder that while linear equations can describe a wide array of lines, the slope-intercept form applies only to those with defined slopes.

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Most popular questions from this chapter

The Apocryphal Manufacturing Company makes widgets out of blivets. If a linear function \(f(x)=m x+b\) gives the number of widgets that can be made from \(x\) blivets, what are the units of the slope \(m\) (widgets per blivet or blivets per widget)?

How do the graphs of \(f(x)\) and \(f(x+10)+10\) differ?

BUSINESS: Break-Even Points and Maximum Profit City and Country Cycles finds that if it sells \(x\) racing bicycles per month, its costs will be \(C(x)=420 x+72,000\) and its revenue will be \(R(x)=-3 x^{2}+1800 x\) (both in dollars). a. Find the store's break-even points. b. Find the number of bicycles that will maximize profit, and the maximum profit.

GENERAL: Water Pressure At a depth of \(d\) feet underwater, the water pressure is \(p(d)=0.45 d+15\) pounds per square inch. Find the pressure at: a. The bottom of a 6 -foot-deep swimming pool. b. The maximum ocean depth of 35,000 feet.

ENVIRONMENTAL SCIENCE: Wind Energy The use of wind power is growing rapidly after a slow start, especially in Europe, where it is seen as an efficient and renewable source of energy. Global wind power generating capacity for the years 1996 to 2008 is given approximately by \(y=0.9 x^{2}-3.9 x+12.4\) thousand megawatts (MW), where \(x\) is the number of years after \(1995 .\) (One megawatt would supply the electrical needs of approximately 100 homes). a. Graph this curve on the window [0,20] by [0,300] . b. Use this curve to predict the global wind power generating capacity in the year \(2015 .\) [Hint: Which \(x\) -value corresponds to \(2015 ?\) Then use TRACE, VALUE, or TABLE.] c. Predict the global wind power generating capacity in the year \(2020 .\)
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