Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither. $$ \begin{array}{l} y=2 x+7 \\ y=-\frac{1}{2} x-4 \end{array} $$

Step-by-Step Solution

Answer

The lines are perpendicular.

Step 1: Identify the Slopes of Both Equations

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the equation \(y = 2x + 7\), the slope \(m\) is 2. For the equation \(y = -\frac{1}{2}x - 4\), the slope \(m\) is -\frac{1}{2}.

Step 2: Determine the Relationship Between the Slopes

Two lines are parallel if their slopes are equal, and they are perpendicular if the product of their slopes is -1. Calculate the product of the slopes: \(2 \times -\frac{1}{2} = -1\). Since the product is -1, the lines are perpendicular.

Step 3: Graph the Equations

To graph the equations, start by plotting the y-intercepts: (0, 7) for the first line and (0, -4) for the second line. From each intercept, use the slope to find another point. For \(y = 2x + 7\), from the point (0, 7), move up 2 units and right 1 unit to locate another point. For \(y = -\frac{1}{2}x - 4\), from the point (0, -4), move down 1 unit and right 2 units to find a second point. Draw the lines through these points.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a line in two variables. It's called that because it tells you both the slope of the line and where it intercepts the y-axis. The general formula is \( y = mx + b \), where:

• \( m \) is the slope, or how steep the line is.
• \( b \) is the y-intercept, or the point where the line crosses the y-axis.

For example, in the equation \( y = 2x + 7 \), the slope \( m \) is 2, and the y-intercept \( b \) is 7. This means the line rises 2 units for every 1 unit it moves horizontally to the right. Similarly, in the equation \( y = -\frac{1}{2}x - 4 \), the slope is \(-\frac{1}{2}\). This slope indicates that the line descends 1 unit for every 2 units it moves to the right. Understanding this form is key to graphing and analyzing linear equations.

Parallel and Perpendicular Lines

Parallel and perpendicular lines have unique relationships in terms of their slopes.

• Parallel lines: Two lines are parallel if they have the same slope. This means they will never intersect because they are "equally steep" and run alongside each other without crossing.
• Perpendicular lines: Lines are perpendicular if the product of their slopes equals -1. This arrangement results in lines that intersect at a right angle (90 degrees).

Given our lines \( y = 2x + 7 \) and \( y = -\frac{1}{2}x - 4 \), their slopes \( 2 \) and \( -\frac{1}{2} \) produce a product of \(-1\). Therefore, these lines are perpendicular. They intersect at a right angle, ensuring a distinct geometric relationship crucial for more advanced geometry topics.

Graphing Techniques

Graphing a linear equation consists of plotting points and drawing a straight line through them based on the slope-intercept form. For each equation, start with the y-intercept:

• For \( y = 2x + 7 \), begin at the point \((0, 7)\).
• For \( y = -\frac{1}{2}x - 4 \), start at \((0, -4)\).

Once the y-intercept is placed:

• Use the slope \( m \) to determine another point.
• For \( m = 2 \), move up 2 and right 1 from \((0, 7)\) to find another point at \((1, 9)\).
• For \( m = -\frac{1}{2} \), move down 1 and right 2 from \((0, -4)\) to \((2, -5)\).

After plotting these points, draw a straight line through them, extending the line beyond these points for clarity. Graphing helps visualize solutions and understand relationships between equations, such as parallelism and perpendicularity.