Problem 48 - Algebra 1: Concepts and Skills

Question

Which are not slopes of perpendicular lines? F. 1 and \(-1\) G. \(-\frac{11}{16}\) and \(\frac{16}{11}\) H. \(\frac{1}{31}\) and \(-31\) J. \(\frac{3}{2}\) and \(\frac{2}{3}\)

Step-by-Step Solution

Answer

Pair J with slopes \(\frac{3}{2}\) and \(\frac{2}{3}\) are not slopes of perpendicular lines.

Step 1: Analyze Pair F

Take the product of 1 and -1. The result is -1, so the lines with these slopes would be perpendicular.

Step 2: Analyze Pair G

Take the product of \(-\frac{11}{16}\) and \(\frac{16}{11}\). The result is -1, so the lines with these slopes would be perpendicular.

Step 3: Analyze Pair H

Take the product of \(\frac{1}{31}\) and -31. The result is -1, so the lines with these slopes would be perpendicular.

Step 4: Analyze Pair J

Take the product of \(\frac{3}{2}\) and \(\frac{2}{3}\). The result is 1, which is not -1. Therefore, the lines with these slopes would not be perpendicular.

Key Concepts

Slope

The concept of slope is central to understanding the geometry of straight lines. In simple terms, slope describes how steep a line is. It is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Denoting this ratio as \( m \), the formula for slope is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of any two distinct points on the line.In the context of perpendicular slopes, it's important to recognize that the slopes of two perpendicular lines multiply together to give -1. This means if the slope of one line is \( m_1 \), and the slope of another line is \( m_2 \), they are perpendicular if: \[m_1 \times m_2 = -1\]Using this criterion, you can determine whether two lines are perpendicular just by looking at their slopes.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only the first powers of the variable. They represent straight lines when graphed on a coordinate plane. The standard form of a linear equation is: \[y = mx + b\]Here, \(m\) is the slope you've learned about, and \(b\) is the y-intercept, where the line crosses the y-axis.These equations are crucial for understanding relationships between variables. When two linear equations are perpendicular, their slopes will multiply to -1, as explained in the prior section. Recognizing this pattern helps determine geometric orientations and angles between lines.

Algebra Basics

Algebra lays the foundation for almost all mathematical concepts you'll explore later. It involves using symbols and letters to represent numbers and express mathematical relationships. Understanding basic algebra is crucial for solving equations and evaluating expressions.In exercises like finding perpendicular lines, it's essential to comprehend how algebra allows us to manipulate equations. You first convert real-world situations into mathematical models, like determining slopes through algebraic expressions.A systematic approach helps when analyzing slopes:

• Substitute the values into the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to find slope.
• Multiply the slopes to check for perpendicularity (product should be -1).

These steps highlight the power of algebra in simplifying complex problems into understandable solutions.