Problem 4 - Algebra 1: Concepts and Skills

Question

Determine whether the lines are perpendicular. $$ y=-4 x+8, y=\frac{1}{4} x+7 $$

Step-by-Step Solution

Answer

Yes, the two lines are perpendicular as the product of their slopes is -1.

Step 1: Identify the Slope for Each Line

The first step in determining if two lines are perpendicular is by identifying their slopes. For the line y = -4x + 8, the slope is -4. For the second line, y = 1/4x + 7, the slope is 1/4.

Step 2: Check the Product of the Slopes

Next you should check what the product of the slopes is. The product of the slopes for these two lines would be -4 * 1/4 = -1.

Step 3: Make a Conclusion

Since the product of the slopes of the two lines is -1, these two lines are perpendicular.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and is a key concept in algebra and geometry. It is represented by the letter 'm' and is calculated as the change in the y-coordinate, known as the rise, over the change in the x-coordinate, known as the run. In simple terms, slope is determined by the ratio \( \frac{\text{rise}}{\text{run}} \).

To find the slope of a line given its equation in the form \( y = mx + b \) where 'm' is the slope and 'b' is the y-intercept, you simply identify the coefficient of the 'x' term. For example, in the line \( y = -4x + 8 \), the slope is -4, indicating the line falls four units vertically for every one unit it moves horizontally. Meanwhile, for the equation \( y = \frac{1}{4}x + 7 \), the slope is \( \frac{1}{4} \), showing a gentle rise of one unit for every four units horizontally.

Product of Slopes

In coordinate geometry, the product of slopes of two lines provides meaningful information about the relationship between the lines. Specifically, when the product of the slopes of two non-vertical lines is -1, the lines are perpendicular to each other. This is because perpendicular lines have slopes that are negative reciprocals of each other.

To find the product of slopes, you simply multiply the slope of one line by the slope of the other. For instance, with the slopes identified in our exercise, -4 and \( \frac{1}{4} \), we calculate their product as \( -4 \times \frac{1}{4} = -1 \). This computation informs us about the relationship between the lines, beyond just the steepness and direction of each individual line. It's a quick test for perpendicularity in Cartesian coordinates.

Perpendicular Lines in Algebra

When two lines are perpendicular to each other in algebra, they intersect at a right angle (90 degrees). This geometric relationship is crucial in various applications, such as in urban planning, construction, and designing objects with orthogonal components.

Algebraically, perpendicular lines have slopes that are negative reciprocals of one another, which means if one line has a slope of 'a', the other line's slope will be \( -\frac{1}{a} \), provided 'a' is nonzero. Such is the case with our example lines: \( y = -4x + 8 \) and \( y = \frac{1}{4}x + 7 \); the slopes are -4 and \( \frac{1}{4} \) respectively. When multiplied together, they result in -1, confirming their perpendicularity algebraically. Understanding this concept allows students to visually and mathematically interpret the orientation of lines with respect to one another.