Question

Give the slope of a line perpendicular to the given line. $$ y=\frac{1}{2} x-3 $$

Short answer

The slope of the line perpendicular to the given line is -2.

Step-by-step solution

Determine the slope of the given line

From the equation of the given line \( y=\frac{1}{2}x -3 \), it can be observed that the slope is \( \frac{1}{2} \).

Find the slope of the perpendicular line

The slope of the line perpendicular to the given line is the negative reciprocal of the slope of the given line. The negative reciprocal of \( \frac{1}{2} \) is \( -\frac{1}{1/2} = -2 \).

Key concepts

Negative Reciprocal

Understanding the concept of a negative reciprocal is essential in algebra, particularly when dealing with perpendicular lines. To find the negative reciprocal of a number, you first flip the fraction, which is the reciprocal part, and then change the sign, which is the negative part.

For instance, if we take a slope of a line as the fraction \( \frac{1}{2} \), the reciprocal would be \( \frac{2}{1} \) or simply 2. To find the negative reciprocal, we then add a negative sign, making it \( -2 \). This is a crucial step when finding the slope of a line that is perpendicular to another, as perpendicular slopes mirror this relationship of being negative reciprocals.

Slope-Intercept Form

The slope-intercept form is a way to describe a line with the equation \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

In our example, the equation \( y = \frac{1}{2}x - 3 \), already presented in the slope-intercept form, makes it easy to identify the slope \( \frac{1}{2} \) and the y-intercept -3. This form is quite useful as it provides a quick way to graph linear equations and understand the rate of change as well as the initial value of the function at a glance.

Linear Equations

Linear equations are fundamental in algebra and represent straight lines when graphed on a coordinate plane. These equations have one or more variables, but each term is either a constant or the product of a constant and a single variable. A key characteristic of linear equations is that they produce a constant rate of change, which can also be interpreted as the slope of the line.

To solve linear equations, one aims to find the values of the variables that make the equation true. Methods to solve linear equations include graphing, substitution, elimination, and using algebraic manipulations. It is through these methods that we can solve for unknowns and understand how different lines interact on a graph, such as intersecting at a point or being parallel or perpendicular to one another.

Perpendicular Lines Algebra

In the realm of algebra, perpendicular lines exhibit a specific relationship: their slopes are negative reciprocals of each other. This means that if two lines are perpendicular, the product of their slopes will be \( -1 \). For instance, if one line has a slope of \( a \), the line perpendicular to it will have a slope of \( -\frac{1}{a} \).

This concept is particularly important when analyzing geometric figures or solving problems involving coordinate geometry. By knowing one line's slope, we can immediately find the slope of a line perpendicular to it, as demonstrated in our initial example. It's a powerful tool that aids in constructing or understanding the behavior of perpendicular lines in various contexts.