Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither. $$ \begin{array}{l} 3 y+4 x=12 \\ -6 y=8 x+1 \end{array} $$

Step-by-Step Solution

Answer

The lines are parallel.

Step 1: Convert Equations to Slope-Intercept Form

Start with the first equation \(3y + 4x = 12\). Solve for \(y\) to put it in the form \(y = mx + b\), where \(m\) is the slope.Subtract \(4x\) from both sides:\[3y = -4x + 12\]Now, divide every term by 3:\[y = -\frac{4}{3}x + 4\]So, the slope of the first line is \(-\frac{4}{3}\).

Step 2: Solve the Second Equation for y

Now, take the second equation, \(-6y = 8x + 1\), and solve for \(y\).First, divide each term by \(-6\):\[ y = -\frac{8}{6}x - \frac{1}{6}\]Simplify the slope:\[y = -\frac{4}{3}x - \frac{1}{6}\]The slope of the second line is also \(-\frac{4}{3}\).

Step 3: Compare the Slopes

The two lines have slopes \(-\frac{4}{3}\) and \(-\frac{4}{3}\), respectively.Since both slopes are equal, this means the lines are parallel.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is a way to express the equation of a straight line. It is represented as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. This form is particularly useful. It allows us to quickly identify the slope and the point at which the line crosses the y-axis. Here are a few key points to keep in mind when using this form:

• Slope \( m \): Indicates the steepness and direction of the line. Positive slopes rise from left to right, while negative slopes fall.
• Y-intercept \( b \): Shows where the line intersects the y-axis.
• Makes it easy to graph by starting at the y-intercept and using the slope to determine the direction and angle of the line.

By converting equations into this form, as seen in the original exercise, it becomes straightforward to determine if two lines are parallel, perpendicular, or neither.

Equations of Lines

Equations of lines can vary in form. Besides the slope-intercept form, equations can also be in the standard form \( Ax + By = C \) or point-slope form \( y - y_1 = m(x - x_1) \). Each form has its specific uses and makes certain calculations simpler. In determining the relationship between two lines, such as parallelism, slope is crucial:

• Parallel lines: Have the same slope but different intercepts.
• Perpendicular lines: Have slopes that are negative reciprocals of each other.
• Neither parallel nor perpendicular: Have slopes that do not fit either condition.

The exercise you encountered required you to find the slope from the given equation. Recognizing the type depends on comparing these slopes effectively.

Algebraic Manipulation

Algebraic manipulation is the process of transforming equations to make them easier to solve or interpret. This can involve moving terms around, factoring, or, as in our original exercise, changing the form of a linear equation. Here’s how it relates to solving for slopes:

• Solving for \( y \): Key to converting a standard form equation to slope-intercept form. Make \( y \) the subject of the equation.
• Maintaining equation balance: Whatever operation you do to one side of the equation (adding, subtracting, multiplying, or dividing), do to the other side.
• Simplifying fractions: Essential for comparing slopes accurately.

By mastering these techniques, you can transform complex equations into simpler, more manageable forms, thereby revealing important line characteristics like slope.