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Chapter 2: Problem 2

Write an equation for the line through (0,50) that has slope: a. -20 b. 5.1 c. 0

Short Answer

Expert verified
y = -20x + 50, y = 5.1x + 50, y = 50

Step by step solution

01

- Understand the slope-intercept form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept.
02

- Identify the given points and slopes

The given point is (0,50), which serves as the y-intercept for all equations. The slopes for parts a, b, and c are -20, 5.1, and 0 respectively.
03

- Write the equation for part a

Using the slope -20 and the y-intercept 50, substitute into the slope-intercept form: \[ y = -20x + 50 \]
04

- Write the equation for part b

Using the slope 5.1 and the y-intercept 50, substitute into the slope-intercept form: \[ y = 5.1x + 50 \]
05

- Write the equation for part c

Using the slope 0 and the y-intercept 50, substitute into the slope-intercept form: \[ y = 0x + 50 \] Simplify the equation to: \[ y = 50 \]

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Key Concepts

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

slope-intercept form
The slope-intercept form is one of the most common ways to write the equation of a line. This form is represented as:
  • \(y = mx + b\)
Where:
  • \(y\) is the dependent variable (usually on the vertical axis)
  • \(x\) is the independent variable (usually on the horizontal axis)
  • \(m\) is the slope of the line
  • \(b\) is the y-intercept of the line
The slope-intercept form is very useful because it directly shows two important characteristics of the line: the slope and the y-intercept. The slope indicates the steepness and direction of the line. The y-intercept tells us where the line crosses the y-axis. We can easily write an equation of a line in this form if we know the slope and y-intercept.
slope
The slope of a line (\(m\)in the equation \(y = mx + b\)) represents how steep the line is. It is calculated as the change in \(y\) divided by the change in \(x\), often written as:
  • \(\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x}\)
A positive slope means the line is ascending from left to right, while a negative slope means it is descending. A zero slope indicates a horizontal line. For example, in the exercise provided:
  • With the slope -20, the line descends steeply.
  • With the slope 5.1, the line ascends moderately.
  • With the slope 0, the line is perfectly horizontal.
Having the slope is essential to determine the direction and angle of the line.
y-intercept
The y-intercept (\(b\)in the equation \(y = mx + b\)) is the point where the line crosses the y-axis. This is where \(x = 0\). It shows the initial value of \(y\) when \(x\) is zero. In the given exercise, the y-intercept is 50. This means all the lines cross the y-axis at the point (0, 50).
  • In part a, the line equation is transformed to \(y = -20x + 50\).
  • In part b, the line equation is changed to \(y = 5.1x + 50\).
  • In part c, the line equation is simplified to \(y = 50\).
Understanding the y-intercept allows us to know exactly where the line will start on the graph, aiding in plotting the line accurately.

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Most popular questions from this chapter

a. If \(S(x)=20,000+1000 x\) describes the annual salary in dollars for a person who has worked for \(x\) years for the Acme Corporation, what is the unit of measure for \(20,000 ?\) For \(1000 ?\) b. Rewrite \(S(x)\) as an equation using only units of measure. c. Evaluate \(S(x)\) for \(x\) values of \(0,5,\) and 10 years. d. How many years will it take for a person to earn an annual salary of \(\$ 43,000 ?\)

The following problems represent calculations of the slopes of different lines. Solve for the variable in each equation. a. \(\frac{150-75}{20-10}=m\) c. \(\frac{182-150}{28-x}=4\) b. \(\frac{70-y}{0-8}=0.5\) d. \(\frac{6-0}{x-10}=0.6\)

Points that lie on the same straight line are said to be collinear. Determine if the following points are collincar. a. \((2,3),(4,7),\) and (8,15) b. \((-3,1),(2,4),\) and (7,8)

Though reliable data about the number of African elephants are hard to come by, it is estimated that there were about 4.000,000 in 1930 and anly 500,000 in 2000 . a. What is the average annual rate of change in elephants over time? Interpret your result b. During the \(1980 \mathrm{~s}\) it was estimated that 100.000 elephants were being killed cach year due to systematic poaching for ivory. How does this compare with your answer in part (a)? What does this tell you about what was happening before or after the \(1980 \mathrm{~s}\) ? (Sounce: ww.panda.org)

The \(y\) -axis, the \(x\) -axis, the line \(x=6,\) and the line \(y=12\) determine the four sides of a 6 -by- 12 rectangle in the first quadrant (where \(x>0\) and \(y>0\) ) of the \(x y\) plane. Imagine that this rectangle is a pool table. There are pockets at the four corners and at the points (0,6) and (6,6) in the middle of each of the longer sides. When a ball bounces off one of the sides of the table, it obeys the "pool rule": The slope of the path after the bounce is the negative of the slope before the bounce. (Hint: It helps to sketch the pool table on a piece of graph paper first.) a. Your pool ball is at (3,8) . You hit it toward the \(y\) -axis, along the line with slope \(2 .\) i. Where does it hit the \(y\) -axis? ii. If the ball is hit hard enough, where does it hit the side of the table next? And after that? And after that? iii. Show that the ball ultimately returns to \((3,8) .\) Would it do this if the slope had been different from \(2 ?\) What is special about the slope 2 for this table? b. A ball at (3,8) is hit toward the \(y\) -axis and bounces off it at \(\left(0, \frac{16}{3}\right) .\) Does it end up in one of the pockets? If so, what are the coordinates of that pocket? c. Your pool ball is at (2,9) . You want to shoot it into the pocket at \((6,0) .\) Unfortunately, there is another ball at (4,4.5) that may be in the way. i. Can you shoot directly into the pocket at (6,0)\(?\) ii. You want to get around the other ball by bouncing yours off the \(y\) -axis. If you hit the \(y\) -axis at \((0,7),\) do you end up in the pocket? Where do you hit the line \(x=6 ?\) iii. If bouncing off the \(y\) -axis at (0,7) didn't work, perhaps there is some point \((0, b)\) on the \(y\) -axis from which the ball would bounce into the pocket at (6,0) Try to find that point.
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