Problem 3 - Algebra 1: Concepts and Skills

Question

Determine whether the lines are perpendicular. $$ y=\frac{1}{5} x-3, y=-5 x+3 $$

Step-by-Step Solution

Answer

Yes, the lines are perpendicular.

Step 1: Find the Slopes of the Lines

The equation of a line in slope-intercept form is \(y = mx + b\), where m is the slope. Thus, for the first line, the slope is \(1/5\) and for the second line, it's \(-5\). We have two slopes, \(m1 = 1/5\) and \(m2 = -5\).

Step 2: Check If the Product of the Slopes is -1

If two lines are perpendicular, then the product of their slopes is -1. Here, the product of the slopes is \((1/5) * (-5) = -1\).

Step 3: Conclusion

Since the product of the slopes is -1, we can conclude that the given lines are perpendicular.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for analyzing the characteristics of a line, including its slope and y-intercept. The slope-intercept form of a linear equation is expressed as
\( y = mx + b \) where \( m \) represents the slope and \( b \) indicates the y-intercept, the point where the line crosses the y-axis. Imagine plotting a line on a graph; the slope dictates the angle and direction of the line, while the y-intercept tells you where to start on the y-axis.

The simplicity of this form allows you to quickly visualize the line. For example, a line with the equation \( y = \frac{1}{5}x - 3 \) has a mild uphill slope moving from left to right, starting at the point (0, -3) on the y-axis. In contrast, \( y = -5x + 3 \) has a steep downhill slope, crossing the y-axis at (0, 3). Visualizing these characteristics can significantly aid in understanding how the lines relate to each other.

Finding Slopes of Lines

The slope of a line measures its steepness, often thought of as 'rise over run.' To find a line's slope in the slope-intercept form \( y = mx + b \) , you simply identify the coefficient of \( x \) as the slope, \( m \).

For the line \( y = \frac{1}{5}x - 3 \) , the slope is \( \frac{1}{5} \) , meaning for every 5 units you run horizontally, you rise by 1 unit. Conversely, for the line \( y = -5x + 3 \) , the slope is \( -5 \) , indicating a sharp descent: for every 1 unit you run, you drop 5 units. Recognizing the slope provides a quick indicator of how the line behaves on a graph without the need for plotting multiple points, and this understanding is crucial when comparing the relationships between lines, especially when determining perpendicularity.

Product of Slopes Condition

When exploring the relationship between two lines, the 'product of slopes' condition is a reliable determinant for perpendicularity. Two lines are perpendicular if and only if the product of their slopes is \( -1 \) .

As an example, let's give consideration to the line with slope \( m_1 = \frac{1}{5} \) and another with slope \( m_2 = -5 \) . The product of these two slopes is \( \frac{1}{5} \times (-5) = -1 \) . This condition fulfills the criteria for perpendicular lines. Therefore, when you encounter two slopes that multiply to \( -1 \) , you can confidently assert that the respective lines are perpendicular to each other. This simple multiplication check is extremely helpful for students to verify perpendicularity without having to graph the lines or delve into more complex geometric proofs.