Problem 14 - Algebra 1: Concepts and Skills

Question

Determine whether the lines are perpendicular. $$ y=\frac{3}{5} x+2, y=-\frac{5}{3} x-2 $$

Step-by-Step Solution

Answer

Yes, the given lines are perpendicular.

Step 1: Identify the slopes

The given equations are \(y= \frac{3}{5}x + 2\) and \(y= -\frac{5}{3}x - 2\). The slopes of the lines are \(\frac{3}{5}\) and \(-\frac{5}{3}\) respectively.

Step 2: Calculate the product of the slopes

Multiply the slopes obtained from step 1. That is, \(\frac{3}{5} \times -\frac{5}{3}\).

Step 3: Compare the product with -1

The multiplication in step 2 results to -1.

Step 4: Conclusion

Since the product of the slopes is -1, the lines are perpendicular according to the condition for perpendicularity of straight lines

Key Concepts

Slope and Its Meaning

In algebra, the concept of slope is incredibly important when analyzing lines on a graph. The slope of a line describes its steepness and direction. It is often denoted as "m" in equations like the slope-intercept form of a line, which is \(y = mx + b\). Here, "m" represents the slope and "b" the y-intercept.
In simple terms, slope can be understood as the change in the vertical direction (y-axis) over the change in the horizontal direction (x-axis) between any two points on the line. This is often referred to as "rise over run."
Let's consider two examples:

• If the slope is positive, like \(\frac{3}{5}\), it means the line is ascending from left to right.
• If the slope is negative, like \(-\frac{5}{3}\), it indicates the line is descending from left to right.

The concept of slope is crucial when determining if lines are perpendicular. For two lines to be perpendicular, the product of their slopes must be \(-1\). This ensures that the lines intersect at a right angle.

Understanding Linear Equations

Linear equations form the backbone of algebraic studies and can be found in various forms. A fundamental form is the slope-intercept form: \(y = mx + b\). Here, "m" is the slope of the line, and "b" is the y-intercept, which tells us where the line crosses the y-axis.
A linear equation gives us a straight line when graphed on a coordinate plane. This means that for every increase in the x-variable, the y-variable will change according to the slope "m." Let's visualize what each part of the equation signifies:

• The coefficient of x ("m") dictates how steep the line is.
• The constant term ("b") provides a starting point for the line on the y-axis.

Linear equations are widely used because they're straightforward to analyze and can model real-world situations effectively. Understanding them helps us solve problems involving network design, budgeting, and even physics.

Core Algebraic Concepts

Algebra involves working with variables, numbers, and operations to solve equations or model scenarios. Several core concepts within algebra help to understand equations better, especially linear equations.
One key concept in algebra is the relationship between variables, often represented in equations. Linear equations, for example, establish a clear, predictable relationship between the variables "x" and "y."
Another crucial aspect is the manipulation of these equations to find unknown values. Solving such equations often involves rearranging them to isolate a particular variable.
Consider the exercise about determining the perpendicularity of lines. To do this, we used these algebraic concepts:

• Extracting the slope from an equation: Recognizing that \(y = \frac{3}{5}x + 2\) means the slope is \(\frac{3}{5}\).
• Performing arithmetic operations with fractions: Calculating products like \(\frac{3}{5} \times -\frac{5}{3}\)
• Applying the perpendicularity condition: Using the slope relationship to confirm that lines are perpendicular by checking if the product of their slopes equals \(-1\).

Mastering these core algebraic skills is invaluable as they build the foundation for more complex mathematics.