Problem 3 - Algebra 1

Question

Give the slope of a line perpendicular to the given line. $$ y=-4 x+2 $$

Step-by-Step Solution

Answer

The slope of the line perpendicular to the given line is \(\frac{1}{4}\).

Step 1: Identify the slope of the given line

The equation of the given line is in the form \(y = mx + n\), where \(m\) is the slope. So, the slope \(m\) of the given line is -4.

Step 2: Calculate the slope of the perpendicular line

The slope of the line perpendicular to the given line would be the negative reciprocal of the given line's slope. The negative reciprocal of -4 is calculated as follows: flip the fraction (4 becoming \(\frac{1}{4}\)) and change the sign (positive to negative or negative to positive, so -4 becomes \(\frac{1}{4}\)). Hence, the slope of the line perpendicular to the given line is \(\frac{1}{4}\).

Step 3: Confirm the Result

To confirm, check if the product of the slope of the given line and the slope of the line perpendicular is -1. The product is (-4) * (\(\frac{1}{4}\)) = -1, so the result is correct.

Key Concepts

Negative Reciprocal

When you hear about the **negative reciprocal**, it may sound complicated, but it's actually very straightforward. Understanding the negative reciprocal is crucial when dealing with perpendicular lines.

To find the negative reciprocal of a number:

• Take its reciprocal. For instance, the reciprocal of 4 is \(\frac{1}{4}\).
• Change its sign. If it's positive, make it negative and vice-versa. Therefore, the negative reciprocal of 4 is \(-\frac{1}{4}\).

This concept helps when dealing with slopes of perpendicular lines, ensuring their product equals \(-1\). Perpendicular slopes always satisfy this condition.

For example, the given line has a slope of \(-4\). Its negative reciprocal would be \(\frac{1}{4}\), making these two lines perpendicular.

Slope of a Line

The **slope of a line** is a measure of its steepness and direction. You can determine how a line slants by calculating its slope using the formula:

\[slope (m) = \frac{\text{rise}}{\text{run}}\text{ or } \frac{y_2 - y_1}{x_2 - x_1}\]In simpler terms:

• The slope tells how much the line goes up or down vertically (rise) for each unit it moves horizontally (run).
• A positive slope goes upward, and a negative slope goes downward.

For example, the line equation \(y = -4x + 2\) is in the slope-intercept form \(y = mx + c\), where \(m\) is the slope. Here, the slope is \(-4\). It tells us that for every unit move to the right, the line goes 4 units downward.

Equation of a Line

The **equation of a line** shows a relationship between the x and y coordinates on a graph. It's typically expressed in the slope-intercept form:\[y = mx + c\]

• \(m\) is known as the slope.
• \(c\) is the y-intercept; where the line crosses the y-axis.

Using the line equation \(y = -4x + 2\):- **Slope (m)**: This is the rate at which y changes with respect to x. Here, it's \(-4\), indicating the line is steep and decreases as it moves to the right.- **Y-intercept (c)**: It's 2, showing that when x is 0, y is 2.This equation tells us every point on the line. Understanding how to read these parts will help in sketching graphs, solving geometric problems, and understanding intersections with lines. Knowing the equation of a line is essential for solving many algebraic tasks that involve coordinates.