Problem 13 - Algebra 1: Concepts and Skills

Question

Determine whether the lines are perpendicular. $$ y=\frac{1}{2} x-7, y=-2 x $$

Step-by-Step Solution

Answer

Yes, the lines are perpendicular.

Step 1: Identify the Slopes

Identify the slopes of the lines. The given equations are: \( y=\frac{1}{2}x - 7 \) and \( y=-2x \). The slopes of these equations are the coefficients of \( x \), which for these equations are \( \frac{1}{2} \) and \( -2 \), respectively.

Step 2: Calculate the Product of the Slopes

Calculate the product of the slopes, which is \(\frac{1}{2} \times -2\). The product equals -1.

Step 3: Determine Perpendicularity

Since the product of the slopes is -1, it can be concluded that the lines are perpendicular as the condition for two lines to be perpendicular is that the product of their slopes is -1.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness. It is an important concept in understanding linear equations. In a linear equation of the form \( y = mx + b \), \( m \) represents the slope. The slope determines how much \( y \) changes for a change in \( x \).

This change in \( y \) with respect to \( x \) is technically known as "rise over run." If you think of a graph, "rise" refers to the vertical change and "run" refers to the horizontal change.

• When \( m > 0 \), the line rises from left to right.
• When \( m < 0 \), the line falls from left to right.
• When \( m = 0 \), the line is horizontal.
• When a line is vertical, its slope is undefined.

In the equation \( y = \frac{1}{2}x - 7 \), the slope is \( \frac{1}{2} \). This means for every unit increase in \( x \), \( y \) increases by half a unit. Conversely, for \( y = -2x \), the slope is \( -2 \), indicating for every unit increase in \( x \), \( y \) decreases by 2 units.

Linear Equations

Linear equations are fundamental mathematical expressions that describe a straight line. These equations can be written in various forms, with the slope-intercept form \( y = mx + b \) being the most common. Here, \( m \) indicates the slope and \( b \) represents the y-intercept, the point where the line crosses the y-axis.

There are a few forms that a linear equation can take:

• Slope-Intercept Form: \( y = mx + b \)
• Point-Slope Form: \( y - y_1 = m(x - x_1) \)
• Standard Form: \( Ax + By = C \)

Understanding these forms enables easier manipulation and solving of linear problems. Linear equations are also essential for identifying relationships between variables, finding slopes, and constructing graphs, all of which are pivotal in analyzing real-world data.

For the original exercise, we used the slope-intercept form to easily identify the slope of each line for determining perpendicularity.

Properties of Perpendicular Lines

Perpendicular lines intersect at a right angle, specifically at 90 degrees. One of their key mathematical properties is related to their slopes. When two lines are perpendicular, the product of their slopes equals -1.

To understand why this is the case, consider two linear equations, \( y = m_1x + b_1 \) and \( y = m_2x + b_2 \). For these lines to be perpendicular, the following must hold: \( m_1 \cdot m_2 = -1 \).

• If the slope of one line is a fraction like \( \frac{1}{2} \), the perpendicular line must have a slope of \(-2\).
• Similarly, if a line has a negative slope such as \(-3\), its perpendicular counterpart will have a slope of \( \frac{1}{3} \).

In the exercise, by identifying that the slopes \( \frac{1}{2} \) and \( -2 \) multiplied to give \(-1\), it confirmed that the two lines were indeed perpendicular. This property is crucial in geometry and helps to determine right angles efficiently.