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Chapter 2: Problem 1

Find the slope of a straight line that goes through: a. (-5,-6) and (2,3) b. (-5,6) and (2,-3)

Short Answer

Expert verified
a. \( \frac{9}{7} \) b. \( -\frac{9}{7} \)

Step by step solution

01

Identify the coordinates

First, identify the given points. For part a, the points are \((-5,-6)\) and \(2,3)\). For part b, the points are \((-5,6)\) and \(2,-3)\).
02

Use the slope formula

Use the formula for the slope of a line: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
03

Calculate the slope for part a

Substitute the coordinates of the points in part a into the slope formula: \(m = \frac{3 - (-6)}{2 - (-5)} = \frac{3 + 6}{2 + 5} = \frac{9}{7}\).
04

Calculate the slope for part b

Substitute the coordinates of the points in part b into the slope formula: \(m = \frac{-3 - 6}{2 - (-5)} = \frac{-3 - 6}{2 + 5} = \frac{-9}{7}\).

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Key Concepts

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

Linear Equations
A linear equation is one that forms a straight line when graphed on a coordinate plane. These equations are in the form of \( y = mx + b \). In this equation, \( y \) is the dependent variable (the outcome), \( x \) is the independent variable (the input), \( m \) is the slope, and \( b \) is the y-intercept, where the line crosses the y-axis.
The key to solving linear equations is understanding their components:
  • Slope (\(m\)) - measures the steepness and direction of a line.
  • Y-intercept (\(b\)) - where the line crosses the y-axis.
Let's dive deeper into each component to understand how linear equations work.
Coordinate Geometry
Coordinate geometry, or analytic geometry, involves placing geometric figures into a coordinate plane. Here's a quick rundown:
Each point on the plane is defined by a pair of numerical coordinates represented as \((x, y)\), which show its position relative to the origin (0,0). The x-coordinate represents the position along the horizontal axis, while the y-coordinate indicates the vertical position.
  • The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
  • The midpoint between two points is found using the midpoint formula: \( \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).
Coordinate geometry helps in visualizing linear equations, which leads us to our next topic: calculating the slope of a line.
Slope Formula
The slope formula is essential for finding the slope of a line given two points. The formula is: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Let's break it down:
  • \( y_2 - y_1 \) - represents the vertical change (rise).
  • \( x_2 - x_1 \) - represents the horizontal change (run).
This ratio defines how steep the line is. For example, if the slope \( m \) is positive, the line ascends from left to right; if it's negative, the line descends. Here鈥檚 how to apply the formula using an example:

Given two points: \( (-5,-6) \) and \( (2,3) \):
  • Identify the coordinates: \( x_1 = -5, y_1 = -6, x_2 = 2, y_2 = 3 \)
  • Substitute them into the slope formula: \( m = \frac{3 - (-6)}{2 - (-5)} = \frac{3 + 6}{2 + 5} = \frac{9}{7} \)
The slope of the line passing through these points is \( \frac{9}{7} \). These simple steps will help you determine the slope for any pair of points.

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Most popular questions from this chapter

(Graphing program optional.) Suppose that: For 8 years of education, the mean annual earnings for women working full-time are approximately \(\$ 19,190\). For 12 years of education, the mean annual earnings for women working full- time are approximately \(\$ 31,190\). For 16 years of education, the mean annual earnings for women working full-time are approximately \(\$ 43,190\). a. Plot this information on a graph. b. What sort of relationship does this information suggest between earnings and education for women? Justify your answer. c. Generate an equation that could be used to model the data from the limited information given (letting \(E=\) years of education and \(M=\) mean earnings). Show your work.

Over a 5 -month period at Acadia National Park in Maine, the average night temperature increased on average 5 degrees Fahrenheit per month. If the initial temperature is 25 degrees, create a formula for the night temperature \(N\) for month \(t,\) where \(0 \leq t \leq 4\)

Find an equation, generate a small table of solutions, and sketch the graph of a line with the indicated attributes. A line that has a vertical intercept of -2 and a slope of 3 .

The accompanying data show rounded average values for blood alcohol concentration \((\mathrm{BAC})\) for people of different weights, according to how many drinks ( 5 oz wine, 1.25 oz 80 -proof liquor, or 12 oz beer) they have consumed. $$ \begin{array}{cccc} \hline \text { Number of Drinks } & 100 \mathrm{lb} & 140 \mathrm{lb} & 180 \mathrm{lb} \\ \hline 2 & 0.075 & 0.054 & 0.042 \\ 4 & 0.150 & 0.107 & 0.083 \\ 6 & 0.225 & 0.161 & 0.125 \\ 8 & 0.300 & 0.214 & 0.167 \\ 10 & 0.375 & 0.268 & 0.208 \\ \hline \end{array} $$ a. Examine the data on BAC for a 100 -pound person. Are the data linear? If so, find a formula to express blood alcohol concentration, \(A,\) as a function of the number of drinks, \(D,\) for a 100 -pound person. b. Examine the data on BAC for a 140 -pound person. Are the data linear? If they're not precisely linear, what might be a reasonable estimate for the average rate of change of blood alcohol concentration, \(A,\) with respect to number of drinks, \(D ?\) Find a formula to estimate blood alcohol concentration, \(A,\) as a function of number of drinks, \(D,\) for a 140 -pound person. Can you make any general conclusions about BAC as a function of number of drinks for all of the weight categories? c. Examine the data on \(\mathrm{BAC}\) for people who consume two drinks. Are the data linear? If so, find a formula to express blood alcohol concentration, \(A,\) as a function of weight, \(W,\) for people who consume two drinks. Can you make any general conclusions about \(\mathrm{BAC}\) as a function of weight for any particular number of drinks?

a. According to the U.S. Bureau of the Census, between 1980 and 2004 domestic new car sales declined from 6581 thousand cars to 5357 thousand. (Note: This does not include trucks, vans, or \(\$ \mathrm{UV}\) s.) Calculate the annual averagc rate of changc. b. During the same period Japanese car sales in the United States dropped from 1906 thousand to 798 thousand. Calculate the average rate of change c. What do the two rates suggest about car sales in the United States?
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