Problem 34 - Algebra Form and Function

Question

Are the lines parallel? $$ y=12+a x ; y=20+a x, \text { where } a \text { is a constant } $$

Step-by-Step Solution

Answer

Answer: Yes, they are parallel.

Step 1: Identify the slopes and y-intercepts of the given lines

The given lines are in the slope-intercept form (y = mx + b), where m is the slope, and b is the y-intercept. For the first line, y = 12 + a x, the slope is 'a' and the y-intercept is 12. For the second line, y = 20 + a x, the slope is 'a' and the y-intercept is 20.

Step 2: Compare the slopes and y-intercepts

To determine if the lines are parallel, we need to check if they have the same slope and different y-intercepts. Both lines have the slope 'a,' and the y-intercepts are different (12 and 20).

Step 3: Conclusion

Since both lines have the same slope ('a') and different y-intercepts (12 and 20), they are parallel.

Key Concepts

Slope

When we talk about slopes in the context of lines on a graph, it is much like describing how steep a hill is. If a line is flat, the slope is 0, and if it goes upwards towards the right, it is positive. A downward, right-leaning slope is negative. You can think of slope as a measurement of how much the line rises vertically for every step it makes horizontally. Thus, it's often referred to as "rise over run."
For the mathematical representation, the slope is often given the symbol \( m \). In the equation of a line, it is the coefficient of \( x \), hence in our example, both lines have a slope of \( a \).

• A larger slope value means a steeper line.
• Slope is constant for straight lines, meaning it doesn't change regardless of where you check it on the line.
• Parallel lines always have the same slope, which is why in our exercise, both lines have slope \( a \), making them parallel.

Y-intercept

The y-intercept is an important part of a line, indicating where the line crosses the y-axis. It's the point where \( x = 0 \). In simpler terms, it's where the line touches or cuts the y-axis when you graph it.
It gives a starting point to plot the line on a graph before considering its slope.In our exercise:

• The first line \( y = 12 + a x \) has a y-intercept of 12.
• The second line \( y = 20 + a x \) has a y-intercept of 20.

The difference in y-intercepts, while having the same slope, confirms the lines are parallel but not coinciding, as they are vertically shifted apart by a difference of 8 units.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that it is immediately apparent what the slope and y-intercept are. This form looks like \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.Using this form makes it easy to quickly identify these two important characteristics of a line, which can help us determine relationships between lines, such as whether they are parallel.

• It is a straightforward method for graphing lines since you always start at the y-intercept.
• The slope tells you how to move from that starting point.
• When comparing lines to check for parallelism, look at the slopes, and if they're equal and the y-intercepts are different, the lines are parallel.

In our exercise, both equations \( y = 12 + ax \) and \( y = 20 + ax \) clearly exhibit the slope-intercept form, having consistent slopes with differing intercepts, indicative of parallel lines.