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Chapter 2: Problem 77

Find an equation of each line. Vertical line through \((1-10, -92)\)

Short Answer

Expert verified
The equation of the vertical line is \(x = -9\).

Step by step solution

01

Title - Understand the Characteristics of a Vertical Line

Recall that a vertical line has an undefined slope and runs parallel to the y-axis. It has the equation of the form \(x = a\), where \(a\) is a constant.
02

Title - Identify the x-coordinate of the given point

Examine the coordinates given. The point is \((1-10, -92)\). Despite the appearance, \(1-10\) simplifies to \(-9\). Therefore, the x-coordinate of the point is \(-9\).
03

Title - Write the Equation of the Vertical Line

Since the x-coordinate of the given point is \(-9\), the equation of the vertical line passing through this point is \(x = -9\). Vertical lines do not depend on the y-coordinate.

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Key Concepts

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

undefined slope
When we talk about the slope of a line, we usually mean how steep the line is. However, in some cases, a line might not have a defined slope. Specifically, vertical lines have what we call an 'undefined slope'. This is because slope is calculated as the ratio of the change in y-coordinates to the change in x-coordinates (rise over run). For a vertical line, there is no change in x-coordinates, which means the denominator in the slope formula is zero. As division by zero is undefined in mathematics, we say the slope of a vertical line cannot be determined. It's important to remember this when dealing with vertical lines.
line equations
Line equations come in different forms depending on the type of line. A vertical line is parallel to the y-axis and has an equation of the form \(x = a\), where \(a\) is a constant representing the x-coordinate of every point on the line. In this exercise, the vertical line passes through the point \((-9, -92)\).
Unlike horizontal or diagonal lines, a vertical line鈥檚 x-value remains constant, and the y-value can be any number. This causes its slope to be undefined. Understanding these different forms of line equations will help you quickly identify and write the equation for any line you encounter.
coordinate geometry
Coordinate geometry, or analytic geometry, allows us to represent geometric figures through algebraic equations and vice versa. It involves points, lines, and figures on the coordinate plane.
In coordinate geometry, each point is defined by an \(x\)-coordinate and a \(y\)-coordinate. The given point in the exercise, \((1-10, -92)\), needs to be simplified to identify the correct coordinates. Simplifying \(1-10\) gives \(-9\).
Therefore, the point is \((-9, -92)\). This point helps us write the line equation for the vertical line. Coordinate geometry is essential in solving various geometric problems by leveraging algebraic methods.

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Most popular questions from this chapter

Solve each equation graphically. Then check your answer by solving the same equation algebraically. $$ \frac{1}{2} x+3=5 $$

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ y(3-x)=2 $$

Find the intercepts. Then graph by using the intercepts, if possible, and a third point as a check. $$ g(x)=2 x-9 $$

Indicate which of the following equations have \(\left(-\frac{1}{3}, \frac{1}{4}\right)\) as a solution. a) \(-\frac{3}{2} x-3 y=-\frac{1}{4}\) b) \(8 y-15 x=\frac{7}{2}\) c) \(0.16 y=-0.09 x+0.1\) d) \(2(-y+2)-\frac{1}{4}(3 x-1)=4\)

Solve each equation graphically. Then check your answer by solving the same equation algebraically. $$ x+2=3 $$
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