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Chapter 3: Problem 66

What are the \(y\) -intercepts of the line \(x=0\) ?

Short Answer

Expert verified
Infinite y-intercepts; line lies on y-axis.

Step by step solution

01

Understanding the Problem

The exercise asks us to find the y-intercepts of a line described by the equation \(x=0\). A y-intercept is the point where the line crosses the y-axis.
02

Analyzing the Equation

We analyze the equation \(x=0\). This equation represents a vertical line in a coordinate plane that is always on the y-axis.
03

Identify the y-intercepts

Since the line \(x=0\) is on the y-axis, it crosses the y-axis at every point along it. This means for every value of \(y\), there is a point on the line.
04

Conclusion

Therefore, the line \(x=0\) has infinitely many y-intercepts because it lies directly on the y-axis.

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Key Concepts

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

Vertical Line
A vertical line in geometry is a straight line that moves up and down without any deviation. In a coordinate plane, a vertical line is represented by the equation \(x = a\), where \(a\) is a constant. This equation tells us that no matter what value \(y\) takes, \(x\) will always be \(a\). Vertical lines are unique because they do not have a slope in the traditional sense, as the slope formula results in division by zero. They are characterized by their vertical orientation and their infinite number of points along the y-axis for the given \(x\)-value.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can graph points, lines, and curves. It is defined by two perpendicular axes: the horizontal x-axis and the vertical y-axis. Together, these axes create a grid where each point is identified by an ordered pair \((x, y)\).
  • The origin, \((0,0)\), is where the x-axis and y-axis intersect.
  • Horizontal movement is determined by the x-coordinate, while vertical movement is determined by the y-coordinate.
The coordinate plane allows us to visually represent algebraic equations and understand geometric relationships. By analyzing equations on this plane, we can easily see how lines and curves behave, including how they intercept the axes.
Equation of a Line
The equation of a line is a fundamental concept that describes all points along a line on the coordinate plane. Commonly, a line is expressed in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. However, vertical lines are a special case. They are represented by an equation like \(x = a\), highlighting their unique properties:
  • Vertical lines do not have a slope, as any attempt to calculate it results in division by zero.
  • The single x-value is the defining feature, indicating that all points on the line share the same x-coordinate.
Understanding the equation of a line helps us graph these lines and analyze their interactions with other geometric figures on the coordinate plane.

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Most popular questions from this chapter

Find the equation of a line, given the slope and a point on the line. $$ m=1 / 2 ;(-4,8) $$

Given the following, \(f(x)=9 x-4,\) find \(x\) when \(f(x)=0 .\)

Is the ordered pair a solution to the given inequality? $$ 6 x-2 y \leq 1 ;(-3,-7) $$

Determine whether the given point is a solution. $$ y=34 x+1 ;(-23,12) $$

Find the intercepts and graph them. $$ 12 x-13 y=1 $$
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