Problem 76 - 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. \(L_{1}: x-4 y-12=0 ; L_{2}: 3 x-4 y-8=0\)

Step-by-Step Solution

Answer

The given lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular.

Step 1: Rewrite Equations into Slope-Intercept Form

The first step is to rearrange the equations into slope-intercept form (i.e. \(y = mx + c\)), where m is the slope. For \(L_{1}: x-4y-12 = 0\), can be rewritten as: \(4y=x-12\), which simplifies to \(y = 1/4x - 3\). Similarly, for \( L_{2}: 3x-4y-8 = 0\), can be rewritten as: \(4y = 3x - 8\), which simplifies to \(y = 3/4x - 2\).

Step 2: Compare the Slopes

The slopes of the lines can be obtained from the \(m\) term in the slope-intercept form. For \(L_{1}\), the slope is 1/4, and for \(L_{2}\), the slope is 3/4.

Step 3: Determine the Relationship Between the Lines

As the slopes of the two lines are not equal, they are not parallel. We must now check if they are perpendicular with the following condition of slopes of two lines being multiplied equals -1. Here, multiplying 1/4 and 3/4 gives 3/16, which is not -1, therefore the lines are not perpendicular.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential for identifying the characteristics of a line. This form is given as \(y = mx + c\), where:

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

To convert a line's equation into this form, solve for \(y\). This makes it simple to identify the slope. For example, consider the equation \(x - 4y - 12 = 0\). To rewrite it, bring x and the constant term to the right side of the equation:

• Step 1: Move \(x\) to the right to isolate the \(y\)-terms: \(4y = x - 12\).
• Step 2: Divide by 4 to solve for \(y\): \(y = \frac{1}{4}x - 3\).

Now the equation is in slope-intercept form, revealing the slope and the y-intercept clearly.

Equation of a Line

The equation of a line in slope-intercept form allows for a comprehensible approach to line comparison. For lines \(L_1\) and \(L_2\), both originally given in standard form, it is crucial to translate these into the more user-friendly slope-intercept form. This transformation aids in analyzing and comparing line equations effectively. By rearranging the equation \(L_1: x - 4y - 12 = 0\) into \(y = \frac{1}{4}x - 3\), and \(L_2: 3x - 4y - 8 = 0\) into \(y = \frac{3}{4}x - 2\), both lines now display their slopes and intercepts. Notice how these final forms parallel the slope-intercept template, making it straightforward to spot similarities and differences. This method demystifies finding relationships between multiple lines, without grappling with more complex calculations. Such clarity is beneficial when comparing characteristics such as parallelism, perpendicularity, or each line's intersection behavior with the coordinate axes.

Slopes of Lines

Slopes are fundamental to understanding line orientation and behavior. The slope is a measure of a line’s steepness and direction. It describes how much y increases as x increases by one unit. The two key relationships to remember when using slopes to determine line relationships are:

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that multiply to -1.

For line \(L_1\), the slope is \(\frac{1}{4}\), and for \(L_2\), the slope is \(\frac{3}{4}\). Since these slopes are different, the lines are not parallel. To explore perpendicularity, multiply the slopes: \(\frac{1}{4} \times \frac{3}{4} = \frac{3}{16}\). Since \(\frac{3}{16} eq -1\), the lines are also not perpendicular. In conclusion, line slopes are a simple yet powerful tool for determining geometrical relationships among lines, ensuring a clear understanding of whether lines run similarly or oppositely on a plane.