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Chapter 3: Problem 3

Find the slope of the line that passes through the given points. See Examples 1 and 2. $$ (1,4) \text { and }(5,3) $$

Short Answer

Expert verified
The slope of the line is \(-\frac{1}{4}\).

Step by step solution

01

Identify the formula for finding slope

To find the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]where \(m\) is the slope of the line.
02

Substitute the coordinates into the formula

Plug the given coordinates \((x_1, y_1) = (1, 4)\) and \((x_2, y_2) = (5, 3)\) into the slope formula:\[m = \frac{3 - 4}{5 - 1}\]
03

Calculate the difference in coordinates

Calculate the numerator \(3 - 4 = -1\) and the denominator \(5 - 1 = 4\).
04

Compute the slope

Using the differences calculated, find the slope by dividing:\[m = \frac{-1}{4}\]The slope of the line is \(-\frac{1}{4}\).

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Key Concepts

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

Linear Equations
Linear equations form the backbone of algebra and are essential for understanding many mathematical concepts. A linear equation represents a straight line when plotted on a coordinate graph. It typically takes the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. This equation is called "linear" because its graph is a straight line.

Understanding the components:
  • The slope (\(m\)) indicates how steep the line is and in which direction it points. A positive slope means the line rises as you move from left to right, while a negative slope indicates it falls.
  • The y-intercept (\(b\)) is where the line crosses the y-axis. This is the value of \(y\) when \(x = 0\).
Linear equations are used extensively in various fields like physics, economics, and engineering to model relationships between variables. They are easy to solve, predict outcomes, and graph, making them a versatile tool in both academic and practical scenarios.
Coordinate Geometry
Coordinate geometry, also known as analytical geometry, is a branch of mathematics that uses algebraic formulas to describe geometric figures and their properties. It allows us to find the slope of a line, which is a measure of how much the line inclines from the horizontal.

Geometry in a coordinate plane:
  • Points are defined by pairs of numbers in an ordered fashion, such as \((x, y)\), which represent their horizontal and vertical positions.
  • The distance and direction between points can be calculated, determining lines, segments, and angles.
  • By using this system, you can solve geometric problems using algebraic methods, which creates a powerful linkage between algebra and geometry.
Coordinate geometry allows us to visualize algebraic equations and offers a visual representation of relationships, providing a clearer understanding of concepts.
Algebraic Formulae
Algebraic formulae are tools that simplify the process of solving equations and problems in mathematics. They provide a structured way to carry out calculations with specific steps and rules. In finding the slope, the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\] is an algebraic method used to quantify the steepness of a line passing through two points.

Breaking down the formula:
  • \(y_2 - y_1\) represents the change in the vertical direction between two points 鈥� known as the "rise."
  • \(x_2 - x_1\) indicates the change in the horizontal direction 鈥� called the "run."
  • Dividing the "rise" by the "run" (\(\frac{\text{rise}}{\text{run}}\)) gives the slope, expressing how much \(y\) changes for a unit change in \(x\).
Algebraic formulae like the slope formula allow us to make logical progressions and predictions in both theoretical and real-world contexts. They condense complex relationships into manageable calculations, simplifying understanding and application.

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Most popular questions from this chapter

The amount \(y\) of land occupied by farms in the United States (in millions of acres) from 1997 through 2007 is given by \(y=-4 x+967\). In the equation, \(x\) represents the number of years after 1997 . (Source: National Agricultural Statistics Service) a. Complete the table. $$ \begin{array}{|c|c|c|c|} \hline \boldsymbol{x} & 4 & 7 & 10 \\ \hline \boldsymbol{y} & & & \\ \hline \end{array} $$ b. Find the year in which there were approximately 930 million acres of land occupied by farms. (Hint: Find \(x\) when \(y=930\) and round to the nearest whole number.) c. Use the given equation to predict when the land occupied by farms might be 900 million acres. (Use the hint for part b.)

It's the end of the budgeting period for Dennis Fernandes, and he has \(\$ 500\) left in his budget for car rental expenses. He plans to spend this budget on a sales trip throughout southern Texas. He will rent a car that costs \(\$ 30\) per day and \(\$ 0.15\) per mile and he can spend no more than \(\$ 500\). a. Write an inequality describing this situation. Let \(x=\) number of days and let \(y=\) number of miles. b. Graph this inequality below. C. Why is the grid showing quadrant I only?

Solve. See Example 4. The table shows the domestic box office (in billions of dollars) for the U.S. movie industry during the years shown. (Source: Motion Picture Association of America) $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Box Office (in billions of dollars) } \\ \hline 2003 & 9.17 \\ \hline 2004 & 9.22 \\ \hline 2005 & 8.83 \\ \hline 2006 & 9.14 \\ \hline 2007 & 9.63 \\ \hline 2008 & 9.79 \\ \hline \end{array} $$ a. Write this paired data as a set of ordered pairs of the form (year, box office). b. In your own words, write the meaning of the ordered pair (2006,9.14) c. Create a scatter diagram of the paired data. Be sure to label the axes appropriately. d. What trend in the paired data does the scatter diagram show?

Graph each inequality. $$ x \leq-1 $$

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. A rock is dropped from the top of a 400 -foot cliff. After 1 second, the rock is traveling 32 feet per second. After 3 seconds, the rock is traveling 96 feet per second. a. Assume that the relationship between time and speed is linear and write an equation describing this relationship. Use ordered pairs of the form (time, speed). b. Use this equation to determine the speed of the rock 4 seconds after it is dropped.
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