Problem 74 - Algebra 2

Question

REVIEW Which of the following best describes the graph of the equations below? $$ \begin{array}{l}{8 y=2 x+13} \\ {24 y=6 x+13}\end{array} $$ F The lines have the same \(y\) -intercept. G The lines have the same \(x\) -intercept. H The lines are perpendicular. J The lines are parallel.

Step-by-Step Solution

Answer

The lines are parallel.

Step 1: Simplify Equation 1

Start by simplifying the first equation \(8y = 2x + 13\). Divide all terms by 8 to get \(y = \frac{1}{4}x + \frac{13}{8}\). This is now in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Simplify Equation 2

Simplify the second equation \(24y = 6x + 13\). Divide all terms by 24 to get \(y = \frac{1}{4}x + \frac{13}{24}\). Now, this is also in the slope-intercept form \(y = mx + b\).

Step 3: Identify the Slopes and Intercepts

From the simplified forms, both lines have the same slope \(\frac{1}{4}\) but different y-intercepts: \(\frac{13}{8}\) for the first equation and \(\frac{13}{24}\) for the second equation.

Step 4: Analyze for Parallelism

Two lines are parallel if they have the same slope and different y-intercepts. Here, both lines have the same slope of \(\frac{1}{4}\) but different y-intercepts, which indicates that the lines are parallel.

Key Concepts

Parallelism

Parallelism is a fundamental concept in geometry, especially when dealing with linear equations. When two lines are parallel, they never intersect and always maintain the same distance apart. In the context of linear equations, two lines are considered parallel if they have the same slope but different y-intercepts. The slope is a measure of how steep a line is, and having the same slope means the lines rise and run at the same rate.

For example, consider the equations from the exercise:

• The first equation is simplified to: \(y = \frac{1}{4}x + \frac{13}{8}\)
• The second equation becomes: \(y = \frac{1}{4}x + \frac{13}{24}\)

Both have the same slope of \(\frac{1}{4}\), indicating they are parallel. However, their y-intercepts differ, supporting that they will never meet, showcasing true parallelism in a coordinate plane.

Slope-Intercept Form

Understanding the slope-intercept form is crucial for analyzing lines. This form is expressed as \(y = mx + b\), where \(m\) represents the slope and \(b\) stands for the y-intercept.

The slope, \(m\), indicates how steep the line is and the direction it goes. A positive slope means the line ascends, while a negative slope indicates it descends. The y-intercept, \(b\), is where the line crosses the y-axis.

By converting equations into the slope-intercept form, identifying properties like parallelism becomes straightforward.

• For the equations \(8y = 2x + 13\) and \(24y = 6x + 13\), simplifying them into \(y = \frac{1}{4}x + \frac{13}{8}\) and \(y = \frac{1}{4}x + \frac{13}{24}\) respectively, makes it easy to spot their identical slopes and differing y-intercepts.

This powerful tool helps us to quickly analyze and graph linear equations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to describe geometric figures using an algebraic approach. It involves the use of coordinates, typically in a two-dimensional plane, to precisely define the position and properties of geometric shapes.

When analyzing linear equations, such as those in our exercise, coordinate geometry plays a key role. Each line can be represented with coordinates that arise from its equation.

• The line \(y = \frac{1}{4}x + \frac{13}{8}\) can be plotted by selecting any x-value and calculating the corresponding y-value.
• Similarly, \(y = \frac{1}{4}x + \frac{13}{24}\) follows the same process.

This approach not only aids in graphing the equations but also in examining their relationship with one another, such as determining parallelism. Coordinate geometry transforms abstract algebraic equations into tangible visual representations, making analysis and understanding intuitive.

By mastering these concepts, students can better appreciate the connections between algebra and geometry and solve real-world problems efficiently.