Problem 57 - Introductory Algebra for College Students

Question

Make Sense? In Exercises \(57-60\), determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When finding the slope of the line passing through \((-1,5)\) and \((2,-3),\) I must let \(\left(x_{1}, y_{1}\right)\) be \((-1,5)\) and \(\left(x_{2}, y_{2}\right)\) be \((2,-3)\)

Step-by-Step Solution

Answer

The statement does not make sense because the slope calculation is not affected by which of the given points is designated as \((x_{1}, y_{1})\) or \((x_{2}, y_{2})\). The slope is the same regardless of the assignment of the points.

Step 1: Identify the formula for the slope

The formula for finding the slope of a line passing through two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is given as \(m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}}\). This formula is indifferent to which point is designated as \((x_{1}, y_{1})\) and which one as \((x_{2}, y_{2})\). It is because both ways will give the same slope for a line passing through these points.

Step 2: Apply the points to the formula

Applying the provided points to the slope formula: \(m = \frac{-3- 5}{2- (-1)} = \frac{-8}{3}\). Now, if we switch the point assignments around, and let \((x_{1}, y_{1})\) be \((2,-3)\) and \((x_{2}, y_{2})\) be \((-1,5)\), we get \(m = \frac{5- (-3)}{-1 - 2} = \frac{8}{-3} = - \frac{8}{3}\). In both cases, the slope is the same.

Step 3: Evaluate the statement

Since the designation of \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) does not influence the calculation of the slope, it is concluded that the statement 'I must let \((x_{1}, y_{1})\) be \((-1,5)\) and \((x_{2}, y_{2})\) be \((2,-3)\)' does not make sense. Because the slope of the line between two points does not depend on which point is assigned as \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\)

Key Concepts

Slope Formula

Learning about the slope of a line is a fundamental concept in algebra, and mastering it is key to understanding much of higher mathematics. The slope is a measure of how steep a line is, and it's calculated by the change in the vertical direction (rise) divided by the change in the horizontal direction (run). The formula to calculate the slope between two points, \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \), is given as \( m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \).

This formula remains consistent regardless of which point you label as \( (x_{1}, y_{1}) \) and which as \( (x_{2}, y_{2}) \), as the end result will always be the same. This means that the process of finding the slope is commutative, and the designation of the points is not as important as understanding how to apply them correctly into the formula.

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be graphed as straight lines on a coordinate plane, and these lines are described by their slopes and y-intercepts. For example, the standard form of a linear equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.

Understanding the slope of a line helps you to graph linear equations more precisely and interpret the rate of change they represent. The slope is particularly significant because it shows how the value of \( y \) changes as \( x \) increases by 1 unit—a foundational concept in the study of functions and calculus.

Coordinate Points

Coordinate points are an essential part of graphing on the Cartesian plane. Each point is composed of an \( x \) and a \( y \) value written as \( (x, y) \). The first value, \( x \), represents the horizontal distance from the origin (0,0), and the second value, \( y \), represents the vertical distance from the origin.

When finding the slope of a line, you will work with two sets of coordinate points to define the line. It is imperative to be comfortable plotting these points on a graph and comprehending their significance in relation to the slope and overall equation of the line. The clear identification of these points is the starting point for any algebraic problem-solving involving linear relationships.

Algebraic Problem-Solving

Algebraic problem-solving involves using algebraic techniques to solve equations and understand relationships between variables. When applying these methods to find the slope of a line, you begin by identifying the given information, which, in this case, are two points provided in coordinate form. Next, you will plug these points into the slope formula, \( m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \), and simplify to achieve the final result.

It's important to note that accuracy in the algebraic manipulation of the slope formula is crucial. Mistakes such as misplacing negative signs or incorrectly subtracting coordinates can lead to an incorrect slope value. Therefore, attention to detail and diligent practice of these methods are necessary to excel in algebra and beyond.