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Chapter 3: Problem 67

Can the equation of a vertical line be written in point-slope form? Explain.

Short Answer

Expert verified
No, the equation of a vertical line cannot be written in point-slope form because the slope of a vertical line is undefined.

Step by step solution

01

Understand the point-slope form

The point-slope form of a line is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, or steepness, of the line and \((x_1, y_1)\) is a point on the line. This form is typically used when we have the slope of the line and one point on the line.
02

Understand the slope of a vertical line

A vertical line goes straight up and down, which means it has no run, or change in \(x\). This implies that the slope is undefined because in a vertical line, it can be noticed that the denominator of the slope formula, \(m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}\), equals zero. Since division by zero is undefined in mathematics, it leads to an undefined slope.
03

Think about the slope in the point-slope equation

In the point-slope equation, the value of \(m\), the slope, is important because it affects the value of \(y\). If this is undefined, the equation cannot work since we can't multiply something with an undefined value.
04

Final conclusion

Given that the slope is an essential part of the point-slope equation and because a vertical line has an undefined slope, it can be concluded that the equation of a vertical line cannot be written in point-slope form.

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Key Concepts

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

Point-Slope Form
The point-slope form of a linear equation helps us describe a line when we know a specific point on the line and the line's slope. This form is useful for many problems in algebra and geometry.
The formula for point-slope form is:
  • \( y - y_1 = m(x - x_1) \)
The concept involves two important components:
- The slope \( m \), which tells us how tilted the line is.- The point \((x_1, y_1)\) through which the line passes.
When you have these two bits of information, you can easily write the line鈥檚 equation. It's a straightforward way to transform information about a line into an equation. Yet, it is limited by requiring a defined slope, which is where vertical lines show their peculiarities.
Undefined Slope
The notion of an undefined slope arises specifically with vertical lines. When you attempt to calculate the slope of a vertical line, you'll run into issues. Let's see why:
The formula for the slope \( m \) is calculated with:
  • \( m = \frac{ ext{rise}}{ ext{run}} = \frac{ ext{change in } y}{ ext{change in } x} = \frac{\Delta y}{\Delta x} \)
For a vertical line:- The "rise" or change in \( y \) can be any value.- The "run" or change in \( x \) is \( 0 \).
This results in division by zero, which is undefined in mathematics. This is why we can say vertical lines have an undefined slope. Since the slope is a crucial part of equations in the point-slope form, an undefined slope means you cannot use this form for vertical lines.
Equation of a Line
Writing the equation of a line allows us to understand and predict the behavior of the line on a graph. The most common forms include point-slope form and slope-intercept form.
However, for vertical lines, we use:
  • \( x = a \)
Where \( a \) is the x-coordinate of every point on the line.
For example, the vertical line passing through \( x = 3 \) is simply expressed as \( x = 3 \), implying all points on this line have an x-coordinate of 3. This equation doesn鈥檛 involve a slope because it鈥檚 not needed; the nature of the line itself, being vertical, is fully described by a single x-value, showing why and how vertical lines stand apart in linear equations.

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Most popular questions from this chapter

Express each inverse variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 2. (OBJECTIVE 2) \(r\) varies inversely with \(s .\) If \(r=40\) when \(s=10,\) find \(r\) when \(s=15\).

Express each inverse variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 2. (OBJECTIVE 2) \(J\) varies inversely with \(v .\) If \(J=90\) when \(v=5,\) find \(J\) when \(v=45\).

Explain the process used to find the \(x\)- and \(y\)-intercepts of the graph of a line.

What does it mean when we say that an equation in two variables has infinitely many solutions?

Set up a variation equation and solve for the requested value. The current in a circuit varies directly with the voltage and inversely with the resistance. If a current of 4 amperes flows when 36 volts is applied to a 9 -ohm resistance, find the current when the voltage is 42 volts and the resistance is 11 ohms.
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