Problem 42 - Algebra 1: Concepts and Skills

Question

Complete the statement with always, sometimes, or never. The line \(y=2 x+3\) is _____ perpendicular to a line with slope \(-2\)

Step-by-Step Solution

Answer

The line \(y = 2x + 3\) is never perpendicular to a line with slope \(-2\).

Step 1: Understand the slope of the given line

The slope-intercept form of a linear equation is \(y = mx + c\) where \(m\) is the slope of the line. From the equation given, \(y = 2x + 3\), one can infer the slope is \(2\).

Step 2: Find the condition necessary for the lines to be perpendicular

Two lines are perpendicular to each other if the product of their slopes is \(-1\). Employ this fact to check the perpendicularity with a line of slope \(-2\). Multiply the slopes: \(2*(-2) = -4\). This fact proves they are not perpendicular.

Step 3: Conclude the answer

Because we see that the product of the slopes is not \(-1\), it means that the line \(y = 2x + 3\) is never perpendicular to a line with a slope of \(-2\). Therefore, the word to complete the sentence is 'never'.

Key Concepts

Slope of a Line

The slope of a line is a crucial concept in understanding how steep a line is on a Cartesian plane. It is represented by the letter \( m \) in the slope-intercept form of a line equation. The slope measures the change in the \( y \)-coordinate as the \( x \)-coordinate changes. Think of it as the "rise over run," the change in vertical distance corresponding to a change in horizontal distance.

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• A zero slope signifies a horizontal line, indicating no rise or fall.
• If the slope is undefined, the line is vertical, and division by zero occurs because the run is zero.

By examining the slope of a line, you can determine many properties of the line and how it interacts with other lines, such as being parallel or perpendicular.

Slope-Intercept Form

The slope-intercept form is one of the simplest ways to express a linear equation. It is written as \( y = mx + c \), where:

• \( m \) represents the slope of the line.
• \( c \) represents the \( y \)-intercept, which is the point where the line crosses the \( y \)-axis.

This form is particularly useful because it clearly shows the slope, making it easy to determine the steepness and direction of the line. Additionally, whenever you have two different linear equations in slope-intercept form, it's simple to determine their relationship:

• If the slopes are equal, the lines are parallel.
• If the slopes are negative reciprocals of each other, the lines are perpendicular.

This makes the slope-intercept form powerful for quickly analyzing and understanding the nature of lines on a graph.

Linear Equations

Linear equations are equations that graph as straight lines on a coordinate plane. They are fundamental in algebra and serve as a bridge to understanding more complex equations. A linear equation typically looks like \( ax + by = c \), but it can also be rearranged to other forms like the slope-intercept form.

• Solutions to a linear equation are points \((x, y)\) that lie on the line represented by the equation.
• The equation describes all possible coordinates that create a straight line.
• Linear equations with different slopes intersect at a single point unless the equations are parallel and therefore, have no points of intersection or have infinite if they are the same line.

Understanding linear equations allows students to explore concepts such as rate of change, y-intercept, and the interaction between multiple lines. They help in forming the foundation for solving real-world problems involving relationships between quantities.