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Chapter 5: Problem 20

Find the \(x\) - and \(y\) -intercepts of the equation. $$y=4$$

Short Answer

Expert verified
The y-intercept is (0, 4). There is no x-intercept.

Step by step solution

01

Identify the Equation

The given equation is a horizontal line: y = 4.
02

Find the y-intercept

To find the y-intercept, set x = 0 in the equation. Since the equation doesn't depend on x, the y-intercept is simply y = 4. So, the y-intercept is (0, 4).
03

Find the x-intercept

To find the x-intercept, set y = 0 in the equation. In this case, the equation y = 4 means the line never crosses the x-axis, as y is always 4 regardless of x. So, there is no x-intercept.

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Key Concepts

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

x-intercept
To find the x-intercept of an equation, you set the y-value to zero and solve for x. The x-intercept represents the point where a graph crosses the x-axis. Think of it as the 'zero height' point of the graph.
If you're given an equation like y = mx + b, you'd set y to 0 and solve for x:
\(0 = mx + b \rightarrow x = -\frac{b}{m}\)
However, in our specific example, the equation is y = 4. Here, no matter what value of x you choose, y is always 4. So if you set y to 0, you get 0 = 4, which is not possible. That means the line never touches the x-axis, and hence, there is no x-intercept.
y-intercept
The y-intercept is the point where the graph crosses the y-axis. To find it, you set x to 0 and solve for y.
In the equation y = mx + b, you set x to 0:
\(y = m(0) + b \rightarrow y = b\)
In our case, the equation is y = 4. Here, since the equation doesn't contain any x term, it means that y is always 4 no matter what x is. So when x is 0, y is simply 4. Hence, the y-intercept is (0, 4).
In summary, the y-intercept is found directly from the equation by setting x to 0 and seeing where the line hits the y-axis.
horizontal line
A horizontal line is a special type of line where all points have the same y-coordinate. The general equation for a horizontal line is y = k, where k is a constant.
In this case, our equation is y = 4, meaning every point on the line has a y-value of 4. This results in a flat, horizontal line crossing the y-axis at (0, 4).
Horizontal lines have interesting properties:
  • They have a slope of 0 because there is no vertical change as you move along the line.
  • They never cross the x-axis (except when y = 0), hence no x-intercepts (in our case).
Understanding these properties helps simplify the process of finding intercepts for these lines.

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Most popular questions from this chapter

Sketch the graph of the line whose points have \(x\) - and \(y\) -coordinates that are negatives of each other. What would the equation of this line be?

Define \(x\) - and \(y\) -intercepts in two ways: (a) In terms of the graph of an equation (b) In terms of an algebraic solution to the equation

Solve the following problem algebraically. Be sure to label what the variable represents. Lamont has invested \(\$ 1,300\) in a savings account that pays \(4 \%\) annual interest. At what interest rate must an additional \(\$ 800\) be invested to produce \(\$ 100\) per year in interest?

Round off to the nearest hundredth when necessary. A business buys a piece of machinery for 12,800 dollar Suppose that the value of the machinery depreciates linearly to a value of 2,400 dollar in 8 years. (a) Write an equation relating the value \(V\) of the machinery after \(n\) years. [ Hint: The value of the machinery in year 0 is 12,800 dollar (b) Find the value of the machine in year 5 (c) How long will it take for the machine to depreciate to a value of zero?

Write an equation of the line satisfying the given conditions. Line has \(x\) -intercept 5 and \(y\) -intercept 2
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