Question

In Problems 111-114, the equations of two lines are given. Determine whether the lines are parallel, perpendicular, or neither. $$ \begin{array}{l} y=2 x-3 \\ y=2 x+4 \end{array} $$

Short answer

The lines are parallel.

Step-by-step solution

Identify the slopes of both lines

From the equations of lines, identify the slope. The equations are in slope-intercept form ( y = mx + b), where m is the slope. For the given equations: y = 2x - 3 and y = 2x + 4 , the slope of both lines is m = 2 .

Compare the slopes

To determine if lines are parallel, perpendicular, or neither, compare their slopes: - If slopes are equal, the lines are parallel. - If slopes are negative reciprocals of each other (e.g., m_1 = 2 and m_2 = -1/2 ), the lines are perpendicular. - Otherwise, the lines are neither parallel nor perpendicular.

Conclusion

Since both lines have the same slope of m = 2 , the lines are parallel.

Key concepts

slope-intercept form

The slope-intercept form of a linear equation is a key concept in algebra.
It is written as \( y = mx + b \), where \( m \) stands for the slope and \( b \) represents the y-intercept. The y-intercept is the point where the line crosses the y-axis.
The form allows you to easily identify the slope and the y-intercept just by looking at the equation.
In the given exercise, both equations \( y = 2x - 3 \) and \( y = 2x + 4 \) are already in slope-intercept form.
You can immediately see that the slope, \( m \), for both lines is 2.
This makes it simple to compare their slopes and determine their relationship.

slope comparison

Once the slopes of the lines are identified, you can determine their relationship.
If the slopes are the same, the lines are parallel.
Parallel lines never intersect. They run alongside each other at a constant distance apart.
If the slopes are negative reciprocals, then the lines are perpendicular.
Perpendicular lines intersect at a 90-degree angle.
In our example, both lines have the same slope of 2, so we can conclude they are parallel.
This is important as it helps you understand how lines relate to one another in a coordinate plane.

linear equations

Linear equations represent straight lines when graphed on the coordinate plane.
They can be expressed in several forms, but the most common are the slope-intercept form and standard form \( Ax + By = C \).
The given equations, \( y=2x-3 \) and \( y=2x+4 \), are examples of linear equations in slope-intercept form.
The simplicity of linear equations makes them fundamental in understanding more complex algebraic concepts.

• Identify the slope \( m \) and y-intercept \( b \) when the equation is in slope-intercept form.
• Compare slopes to determine if lines are parallel, perpendicular, or neither.
• Utilize linear equations to solve real-world problems involving constant rates and ratios.

Understanding linear equations is crucial for mastering more advanced mathematical topics.