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Chapter 3: Problem 50

What is the common name given to a line with slope 0 whose \(y\) -intercept is the origin?

Short Answer

Expert verified
The x-axis.

Step by step solution

01

Understanding the Linear Equation

Recall that the equation of a line in slope-intercept form is given by: \[ y = mx + b \]. Here, \( m \) is the slope and \( b \) is the y-intercept.
02

Identify the Slope

In this problem, it is stated that the slope is 0. This implies that \( m = 0 \). Thus, the equation simplifies to: \[ y = 0 \times x + b \] or more simply \[ y = b \].
03

Determine the y-intercept

It is given that the y-intercept is the origin. This means \( b = 0 \). Substituting \( b \) in the simplified equation gives us: \[ y = 0 \].
04

Conclusion: Name of the Line

A line where \( y = 0 \) is commonly known as the x-axis.

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Key Concepts

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

slope-intercept form
In the study of linear equations, one of the most important forms is the slope-intercept form. The formula is expressed as: y = mx + bIn this equation:
  • m represents the slope of the line.
  • b represents the y-intercept.
This form is very useful because it directly gives you both the slope and the y-intercept, making it easier to graph the line or understand its properties.For example, if you have a line with equation y = 2x + 3, the slope m is 2 and the y-intercept b is 3. This means the line rises 2 units up for every 1 unit it moves to the right, and it crosses the y-axis at 3.
x-axis
The x-axis is one of the two number lines in the coordinate plane. It is the horizontal line where the value of y is always 0. In other words, every point on the x-axis has a y-coordinate of 0. This is significant when we talk about a line with an equation y = 0. This equation tells us that no matter what value x takes, y will always be 0. Therefore, the line is actually the x-axis itself. So, a line with slope 0 and a y-intercept of the origin is simply the x-axis. It stretches infinitely in both the positive and negative directions along the x-coordinate.
y-intercept
The y-intercept is the point where a line crosses the y-axis. To find it, you look at the constant term b in the slope-intercept form of a line's equation, y = mx + b. The y-intercept occurs when x = 0:
  • If b is positive, the line crosses the y-axis above the origin.
  • If b is negative, the line crosses below the origin.
  • If b is zero, the line intersects the origin itself.
For example, in the given problem, since the slope is 0 and the y-intercept is the origin, b = 0. This translates to a line with the equation y = 0. Hence, this line is the x-axis. Understanding the y-intercept helps in quickly identifying where the line will be positioned relative to the y-axis.

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Most popular questions from this chapter

It costs a flat fee of 20 dollars plus 5 dollars per day to rent a pressure washer. Therefore, the cost to rent the pressure washer for \(x\) days is given by $$ y=5 x+20 $$ where \(y\) is in dollars. Express each of the following as an ordered pair. (a) When the washer is rented for 5 days, the cost is 45 dollars. (b) I paid 50 dollars when I returned the washer, so I must have rented it for 6 days.

Graph each equation by using the slope and y-intercept. $$ 4 x-5 y=20 $$

Write an equation of the line satisfying the given conditions. Give the final answer in slope intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines in Section \(3.3 .)\) Through \((2,3) ; \quad\) parallel to \(4 x-y=-2\)

Suppose that it costs 5000 dollars to start up a business selling snow cones. Furthermore, it costs 0.50 dollars per cone in labor, ice, syrup, and overhead. Then the cost to make \(x\) snow cones is given by \(y\) dollars, where $$ y=0.50 x+5000 $$ Express each of the following as an ordered pair. (a) When 100 snow cones are made, the cost is 5050 dollars. (b) When the cost is 6000 dollars, the number of snow cones made is 2000 .

Concept Check Plot each set of points, and draw a line through them. Then give the equation of the line. $$ (-5,5),(0,5), \text { and }(3,5) $$
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