Problem 48 - Algebra 1

Question

Decide whether the graphs of the two equations are parallel lines. Explain your answer. $$ y+6 x-8=0,2 y=12 x-4 $$

Step-by-Step Solution

Answer

The two equations represent lines that are not parallel, because their slopes are not the same

Step 1: Rewrite the first equation in slope-intercept form

To rewrite the first equation \(y + 6x - 8 = 0\) in slope-intercept form, isolate 'y' by subtracting \(6x\) from both sides and then adding \(8\). This results in \(y = -6x + 8\).

Step 2: Rewrite the second equation in slope-intercept form

To rewrite the second equation \(2y = 12x - 4\) in slope-intercept form, divide both sides by 2 to isolate 'y'. This results in \(y = 6x - 2\).

Step 3: Compare the slopes

From the first equation, the slope is \(-6\), and from the second equation, it is \(6\). Because these slopes are not equal, the two lines are not parallel.

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never meet. They have the same slope but different y-intercepts. Imagine them like railroad tracks extending into the horizon, never converging or diverging. For two equations to describe parallel lines, the slopes of these lines must be exactly the same.

In our exercise, we have two distinct linear equations:

• The first equation is transformed into the form: \(y = -6x + 8\).
• The second equation becomes: \(y = 6x - 2\).

For lines to be parallel, their slopes must be equal. In these equations, one line has a slope of \(-6\), while the other has a slope of \(6\). These slopes being different means these lines will eventually cross each other on the graph. Therefore, they are not parallel.

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations using the formula \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form is handy because it immediately tells us two key things:

• 'm', the slope, indicates how steep the line is.
• 'b', the y-intercept, indicates where the line crosses the y-axis.

To apply this in the task, we start by converting each given equation into this form:- For \(y + 6x - 8 = 0\), isolate \(y\) to get \(y = -6x + 8\).

- For \(2y = 12x - 4\), we divide everything by 2 to simplify it to \(y = 6x - 2\).

Now, by looking at \(m\), we can easily compare the slopes of lines and determine their relationship, such as if they're parallel or not. The slope-intercept form simplifies analyses like these considerably.

Graph Interpretation

Understanding a graph involves recognizing the relationships between different graphical elements, especially when dealing with equations. The slope tells us if a line is increasing or decreasing as we move from left to right. If the slope is positive, as in \(y = 6x - 2\), the line will rise. Conversely, with a negative slope, like in \(y = -6x + 8\), the line falls.

By analyzing the y-intercept, \(b\), you also know where these lines cross the y-axis:

• The line from \(y = -6x + 8\) crosses at (0, 8).
• Meanwhile, the line from \(y = 6x - 2\) crosses at (0, -2).

Graph interpretation allows us to visualize how equations translate into lines. When interpreting these specific graphs, both lines have different starting points and slopes, meaning that they will behave quite differently on the graph, confirming their non-parallel nature. By examining these attributes, we gain a deeper comprehension of their distinct paths on the graph.