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Chapter 2: Problem 29

Find the slope of the line that passes through the given pair of points. $$ (a, b) \text { and }(c, d) $$

Short Answer

Expert verified
The slope of the line passing through the given points \((a, b)\) and \((c, d)\) is: $$ m = \frac{d - b}{c - a} $$

Step by step solution

01

Write down the slope formula

The slope formula, denoted by m, is given by the equation: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ This equation helps in finding the slope of a line by dividing the change in the y-coordinates by the change in the x-coordinates.
02

Substitute the coordinates of the given points

Substitute the coordinates of the given points (a, b) as \((x_1, y_1)\) and (c, d) as \((x_2, y_2)\) in the slope formula. That is: $$ m = \frac{d - b}{c - a} $$
03

Simplify the expression for the slope

Since the given points are (a, b) and (c, d) and we've already substituted them into the slope formula, there are no further steps to simplify the expression. The slope of the line passing through the given points is: $$ m = \frac{d - b}{c - a} $$ This is the final answer, and the expression provided calculates the slope of the line passing through the points (a, b) and (c, d).

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Key Concepts

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

Slope Formula
Understanding the slope of a line is a crucial concept in algebra and coordinate geometry. The slope, typically represented by the letter m, is a measure of how steep a line is. In simpler terms, it tells you how much the line rises or falls as it moves from left to right across a graph.

The slope formula is a handy tool that gives us an algebraic method to calculate the slope of a line when two points on the line are known. This formula is:
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is an area of mathematics that describes the position of points, lines, and shapes within a two-dimensional space. All positions are defined using a pair of numerical coordinates which are placed on a grid composed of perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).

Points are plotted using the notation \( (x, y) \) where \( x \) and \( y \) represent their respective position on the axes. When finding the slope of a line using the slope formula, these coordinates play a key role as they are substituted into the formula to calculate the rate of change from one point to another.
Algebraic Expressions
At the heart of solving many algebraic problems, including finding the slope of a line, lies the ability to manipulate algebraic expressions. These expressions comprise numbers, variables, and operation signs, and they represent values that can vary or be calculated.

When you use the slope formula, you work with an algebraic expression to determine the slope. For instance, after substituting the known point values into the formula, you simplify the expression to arrive at the slope, if possible. Simplification may involve canceling factors, combining like terms, or performing arithmetic operations to reach the most reduced form of the expression.

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Most popular questions from this chapter

A stone is thrown straight up from the roof of an 80 -ft building. The distance of the stone from the ground at any time \(t\) (in seconds) is given by $$ h(t)=-16 t^{2}+64 t+80 $$ a. Sketch the graph of \(h\). b. At what time does the stone reach the highest point? What is the stone's maximum height from the ground?

The immigration to the United States from Europe, as a percentage of the total immigration, is approximately \(P(t)=0.767 t^{3}-0.636 t^{2}-19.17 t+52.7 \quad(0 \leq t \leq 4)\) where \(t\) is measured in decades, with \(t=0\) corresponding to the decade of the \(1950 \mathrm{~s}\). a. Complete the table: b. Use the result of part (a) to sketch the graph of \(P\). c. Use the result of part (b) to estimate the decade when the immigration, as a percentage of the total immigration, was the greatest and the smallest.

According to a study conducted in 2003, the total number of U.S. jobs (in millions) that are projected to leave the country by year \(t\), where \(t=0\) corresponds to the beginning of 2000 , is $$ N(t)=0.0018425(t+5)^{2.5} \quad(0 \leq t \leq 15) $$ What was the projected number of outsourced jobs for 2005 \((t=5) ?\) For \(2010(t=10)\) ?

Patricia's neighbor, Juanita, also wishes to have a rectangular-shaped garden in her backyard. But Juanita wants her garden to have an area of \(250 \mathrm{ft}^{2}\). Letting \(x\) denote the width of the garden, find a function \(f\) in the variable \(x\) giving the length of the fencing required to construct the garden. What is the domain of the function? Hint: Refer to the figure for Exercise 26. The amount of fencing required is equal to the perimeter of the rectangle, which is twice the width plus twice the length of the rectangle.

A book designer has decided that the pages of a book should have 1 -in. margins at the top and bottom and \(\frac{1}{2}\) -in. margins on the sides. She further stipulated that each page should have an area of 50 in. \(^{2}\). Find a function in the variable \(x\), giving the area of the printed page. What is the domain of the function?
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