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Chapter 3: Problem 88

Find the equation of the \(y\) -axis and its slope, if any.

Short Answer

Expert verified
The equation of the y-axis is \(x = 0\) and its slope is undefined.

Step by step solution

01

Identify what the y-axis is in a plane

In a coordinate plane, the y-axis is the vertical line that intersects the x-axis at the origin (0, 0). It represents the transition between negative and positive y-values.
02

Understand the equation of y-axis

The equation of the y-axis is \(x = 0\). This is because at every point on the y-axis, the x-coordinate is 0, regardless of the value of the y-coordinate.
03

Understand the concept of the slope of a line

In geometry, the slope of a line is a measure of how steep the line is. The slope of a line is represented by 'm' in the equation \(y = mx + c\). It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.
04

Determine the slope of the y-axis

The y-axis is a vertical line. For vertical lines, the run (change in x) is 0, as the x-coordinate remains the same. Since the slope is the rise divided by the run, and it's not possible to divide by zero, the slope of the y-axis is considered undefined.

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Key Concepts

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

y-axis
In the coordinate plane, we often talk about two lines that define the plane: the x-axis and the y-axis. The y-axis is a fundamental vertical line that meets the x-axis at point (0,0), known as the origin. Its main role is to represent the transition between negative values (usually below the x-axis) and positive values (above the x-axis). Any point located directly on this vertical line will have its x-coordinate equal to zero, and this is crucial when defining the equation of the y-axis.
equation of a line
In mathematics, lines in a plane can be described using equations. The simplest form is the linear equation. For the y-axis, this equation is particularly simple: it is defined by the equation \(x = 0\). This indicates that no matter where you are on the y-axis, the x-coordinate remains at zero. Unlike many other lines that might have equations like \(y = mx + c\), representing lines with varying slopes and y-intercepts, the y-axis's equation focuses solely on x being zero.
slope of a line
The slope of a line represents its steepness. It's determined by the change in y-values divided by the change in x-values between two points. The slope is commonly symbolized by the letter \(m\) in the equation of the line \(y = mx + c\). However, when it comes to the y-axis, things are different. The y-axis is a vertical line where the x-value does not change (the run is zero). Thus, the division involves zero in the denominator, which leads to an undefined slope, since dividing by zero is mathematically indeterminate. Vertical lines, like the y-axis, always have slopes that are considered undefined.

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Most popular questions from this chapter

Fill in the blanks. Assume that \(k\) is a constant. The equation \(y=k x z\) represents _____ variation.

Set up a variation equation and solve for the requested value. The interest earned on a fixed amount of money varies jointly with the annual interest rate and the time that the money is left on deposit. If an account earns \(120\)dollar at \(8 \%\) annual interest when left on deposit for 2 years, how much interest would be earned in 3 years at an annual rate of \(12 \% ?\)

Set up a variation equation and solve for the requested value. The force of gravity acting on an object varies directly with the mass of the object. The force on a mass of 5 kilograms is 49 newtons. What is the force acting on a mass of 12 kilograms?

Express each joint variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 3. (OBJECTIVE 3) D varies jointly with \(p\) and \(q\). If \(D=16\) when \(p\) and \(q\) are both 8 , find \(D\) when \(p\) and \(q\) are both 12 .

Express each sentence as a formula. For a fixed area, the length \(l\) of a rectangle varies inversely with its width \(w\).
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