Problem 55 - Precalculus

Question

Verify the identity $$\left(y_{2}-y_{1}\right) /\left(x_{2}-x_{1}\right)=\left(y_{1}-y_{2}\right) /\left(x_{1}-x_{2}\right)$$ What does this identity tell you about calculating slope?

Step-by-Step Solution

Answer

The identity shows that reversing the order of points in the slope formula still yields the same slope.

Step 1: Understanding the Expression

The given identity is \[ \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{y_{1}-y_{2}}{x_{1}-x_{2}} \]This identity represents the slope formula rewritten in two different forms for a line that passes through the points \((x_1, y_1)\) and \((x_2, y_2)\).

Step 2: Simplifying Both Sides

We simplify the expression on both sides of the identity:- Left Side: \( \frac{y_{2}-y_{1}}{x_{2}-x_{1}} \) is the standard form of the slope of a line.- Right Side: \( \frac{y_{1}-y_{2}}{x_{1}-x_{2}} \) can be rewritten as \( \frac{-(y_{2}-y_{1})}{-(x_{2}-x_{1})} \), which simplifies to the same slope.

Step 3: Applying Properties of Numbers

Use the property of fractions that states \( \frac{-a}{-b} = \frac{a}{b} \). Since both numerators and denominators on the right side are negated, the negatives cancel each other out, thus confirming that both sides of the identity represent the same value.

Key Concepts

Understanding Coordinates

Coordinates are a crucial concept when discussing any point in the context of a plane or space. A coordinate is an ordered pair \(x, y\) that helps pinpoint a location on the Cartesian coordinate system. It tells us the exact position of a point by defining its horizontal and vertical distances from a set origin, often referred to as the \(0, 0\) point.

Understanding how to work with coordinates is fundamental in solving geometrical and algebraic problems. Each coordinate pair expresses two integers.

• The \(x\) value indicates horizontal position (left or right).
• The \(y\) value specifies vertical position (up or down).

When dealing with two points, such as \( (x_1, y_1) \) and \( (x_2, y_2) \), these coordinates allow us to calculate the slope of the line joining them. This is essential for understanding the geometric layout and relationships between different points on a graph.

Exploring Linear Equations

A linear equation is fundamental in mathematics and describes a straight line when graphed on a coordinate plane. It takes the form \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.

Solving a linear equation involves finding all the points \(x, y\) that satisfy this relationship. These points collectively form a straight line. Recognizing and working with linear equations allows students to handle problems involving consistent rates of change between variables.

The slope of a line, represented by the variable \(m\), indicates the steepness and direction of the line. It is calculated based on the change in \(y\) values divided by the change in \(x\) values between two points:

• Positive slope: Line rises from left to right.
• Negative slope: Line falls from left to right.

The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Understanding how to derive and manipulate linear equations is key to identifying relationships in algebra.

Decoding Algebraic Identity

An algebraic identity is an equation that is true for all values of its variables. It represents a fundamental truth about numbers and their relationships.

The given exercise illustrates an important algebraic identity involving the slope formula: \( \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_1 - y_2}{x_1 - x_2} \). While this may seem complex at first glance, it reveals that both expressions yield the same result due to the properties of subtraction and division:

• Negating both the numerator and the denominator (of one form) leads to the same slope value as reversing the order of subtraction.
• This concept is crucial for ensuring calculations of slope remain consistent regardless of point order.

Grasping the concept of algebraic identities helps students deepen their understanding of algebra basics and offers insights into substituting and simplifying algebraic expressions effectively.