Problem 37 - College Algebra

Question

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither. \(3 x-2 y=5\) \(6 y-9 x=6\)

Step-by-Step Solution

Answer

The lines are parallel.

Step 1: Convert First Equation to Slope-Intercept Form

Start with the first equation given:\[3x - 2y = 5\]To express it in the slope-intercept form \(y = mx + b\), isolate \(y\):Subtract \(3x\) from both sides:\[-2y = -3x + 5\]Divide everything by \(-2\):\[y = \frac{3}{2}x - \frac{5}{2}\]The slope \(m\) of this equation is \(\frac{3}{2}\).

Step 2: Convert Second Equation to Slope-Intercept Form

Start with the second equation:\[6y - 9x = 6\]Convert to slope-intercept form by isolating \(y\):Add \(9x\) to both sides:\[6y = 9x + 6\]Now divide everything by \(6\):\[y = \frac{9}{6}x + 1\]Simplify \(\frac{9}{6}\) to give:\[y = \frac{3}{2}x + 1\]The slope \(m\) for this equation is also \(\frac{3}{2}\).

Step 3: Determine Relationship Between the Lines

Compare the slopes of both equations. The first equation has a slope \(m_1 = \frac{3}{2}\), and the second equation has a slope \(m_2 = \frac{3}{2}\).Since both slopes are equal, the lines are parallel.

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form is a way to express a linear equation as \( y = mx + b \). In this equation, \( m \) represents the slope of the line, while \( b \) is the y-intercept where the line crosses the y-axis. This form is incredibly useful for quickly visualizing and graphing linear equations.

• **Slope \( (m) \):** This indicates the steepness or incline of the line. A positive slope means the line rises as you move from left to right, while a negative slope means it falls.
• **Y-Intercept \( (b) \):** This is the point where the line crosses the y-axis. It gives us a starting point for graphing the line.

To convert a standard form equation (like \( 3x - 2y = 5 \)) into the slope-intercept form, you need to solve for \( y \). This involves rearranging the terms in the equation to isolate \( y \) on one side.

Graphing Linear Equations

Once a linear equation is in slope-intercept form, graphing it becomes straightforward.

• **Identify the slope \( m \):** This tells you how much the line rises or falls as you move along the x-axis.
• **Locate the y-intercept \( b \):** This is your starting point on the graph. Place a point on the y-axis at this value.
• **Use the slope:** From the y-intercept, use the slope to find your next point. If \( m \) is \( \frac{3}{2} \), move up 3 units and right 2 units from the y-intercept to plot your second point.

Draw a line through these two points, extending it through both directions. This visually represents the linear equation on the coordinate plane.

Parallel and Perpendicular Lines in Linear Equations

When comparing linear equations, understanding their geometric relationships is crucial. Lines can be parallel, perpendicular, or neither.

• **Parallel Lines:** These lines have the same slope, which means they rise and fall at the same rate but on different paths. Since they never intersect, the distance between them remains constant.
• **Perpendicular Lines:** These lines intersect at a 90-degree angle. To identify them, look for slopes that are negative reciprocals, such as \( m_1 = \frac{3}{2} \) and \( m_2 = -\frac{2}{3} \).
• **Neither:** If lines are neither parallel nor perpendicular, they intersect in a way that does not form a right angle.

For the given equations, both lines have the same slope of \( \frac{3}{2} \). Thus, they are parallel, as demonstrated by both equations \( y = \frac{3}{2}x - \frac{5}{2} \) and \( y = \frac{3}{2}x + 1 \). This means they never meet on the graph plane.