Problem 73 - College Algebra and Calculus: An Applied Approach

Question

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: x-3 y=-3 ; L_{2}: 2 x-6 y=6\)

Step-by-Step Solution

Answer

The lines are parallel.

Step 1: Rewrite the equations

We need to write \(L_{1}\) and \(L_{2}\) in the form \(y = mx + b\). This gives us \(L_{1}: y = \frac{1}{3}x + 1\) and \(L_{2}: y = \frac{1}{3}x - 1\).

Step 2: Compare the slopes

We see that the slope \(m\) of \(L_{1}\) and \(L_{2}\) is \(\frac{1_{3}\). Thus the slopes are equal.

Step 3: Determine the relationship between the lines

Since the two slopes are equal, it can be deduced that the two lines \(L_{1}\) and \(L_{2}\) are parallel.

Key Concepts

Parallel Lines

Understanding parallel lines in algebra is crucial for solving various geometry and coordinate problems. Parallel lines are defined as lines in the same plane that never intersect, regardless of how far they are extended in either direction. They have the same steepness, or mathematically speaking, the same slope.

In the exercise provided, we determined whether lines \(L_{1}\) and \(L_{2}\) are parallel by comparing their slopes. To achieve this, we first converted the given equations into slope-intercept form, which is \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. Through this process, we found that both lines have an identical slope of \(\frac{1}{3}\), confirming that they are indeed parallel.

Remember, when dealing with potential parallel lines, the required condition is not just having the same slope but also distinct y-intercepts, that is why ensuring the equations are in slope-intercept form is vital. This helps to avoid mistakenly identifying coinciding lines (lines that are on top of each other) as parallel.

Slope of a Line

The slope of a line is a measure of its inclination or steepness, often represented as \(m\) in linear equations. It signifies the rate at which the line rises or falls as you move along it. Calculating the slope involves measuring the vertical change (rise) over the horizontal change (run). The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

The slope is a key element in understanding the nature of the line. A positive slope indicates the line rises as it moves from left to right, while a negative slope suggests it falls. If the slope is zero, the line is horizontal, and if it's undefined (when the run is zero), the line is vertical.

In the provided exercise, the calculation of the slopes allowed us to conclude that the lines are parallel. Knowing how to find and interpret slope is critical for solving and graphing linear equations, and for understanding the relationship between multiple lines in a system.

Linear Equations

Linear equations form the foundation for much of algebra and consist of expressions where the highest power of the variable is one. They can be presented in various forms, including slope-intercept form \(y = mx + b\), point-slope form, and standard form \(Ax + By = C\).

The slope-intercept form, which was used in the exercise to identify parallel lines, showcases two fundamental characteristics of the line: its slope \(m\) and y-intercept \(b\). This form is particularly advantageous for graphing purposes, as it directly provides the rate at which the line ascends or descends (slope) and where it crosses the y-axis (y-intercept).

To tackle linear equations proficiently, it's essential to be comfortable with rearranging them into different forms, solving for variables, and graphing them. The ability to manipulate linear equations is not only key for straight-line analysis but also in solving systems of equations where multiple lines interact.