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Chapter 2: Problem 112

The equations of two lines are given. Determine whether the lines are parallel, perpendicular, or neither. $$ \begin{array}{l} y=\frac{1}{2} x-3 \\ y=-2 x+4 \end{array} $$

Short Answer

Expert verified
The lines are perpendicular.

Step by step solution

01

- Identify the slopes of the lines

For the given lines, convert their equations to slope-intercept form \(y = mx + b\). The equations are already in this form: \(y = \frac{1}{2}x - 3\) and \(y = -2x + 4\). Here, the slopes \(m\) of the lines are \(\frac{1}{2}\) and \(-2\) respectively.
02

- Compare the slopes

Two lines are parallel if their slopes are equal \(m_1 = m_2\). Two lines are perpendicular if the product of their slopes is \(-1\) \(m_1 \times m_2 = -1\).
03

- Determine if lines are parallel, perpendicular, or neither

Check if \(\frac{1}{2} = -2\). Since they are not equal, the lines are not parallel. Next, check whether \(\frac{1}{2} \times -2 = -1\). Since \(\frac{1}{2} \times -2 = -1\), the lines are perpendicular.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line. The general formula for this form is given by:

\( y = mx + b \).

Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept. The y-intercept is the point where the line crosses the y-axis.

For example, in the equation \(y = \frac{1}{2} x - 3\), \( \frac{1}{2} \) is the slope and \(-3\) is the y-intercept. This tells us that the line rises by 1 unit for every 2 units it moves to the right.

Knowing how to identify the slope and y-intercept from a line's equation is crucial for understanding how the line behaves.
Comparing Slopes
Comparing the slopes of two lines helps determine their relationship to each other.

When you have the slopes \(m_1\) and \(m_2\) of two lines, you can decide if the lines are parallel, perpendicular, or neither:

  • **Parallel Lines**: If \(m_1 = m_2\), then the lines are parallel. Parallel lines never intersect and have the same slope.

  • **Perpendicular Lines**: If the product of the slopes \(m_1 \times m_2 = -1\), then the lines are perpendicular. Perpendicular lines intersect at a right angle (90 degrees).

  • **Neither**: If neither of the above conditions is met, the lines are neither parallel nor perpendicular.

In our exercise, we found the slopes \( \frac{1}{2} \) and \(-2\). Since \( \frac{1}{2} eq -2\), the lines are not parallel. And since \( \frac{1}{2} \times (-2) = -1\), the lines are indeed perpendicular.
Perpendicular Lines
Perpendicular lines are lines that intersect at a right angle (90 degrees). The key characteristic of perpendicular lines is the relationship between their slopes.

The slopes of two perpendicular lines are negative reciprocals of each other. This means if one line has a slope of \(m\), the slope of the line perpendicular to it will be \( -\frac{1}{m} \).

For example, if one line has a slope of \( \frac{1}{2} \), a line perpendicular to it will have a slope of \( -2 \) because \( \frac{1}{2} \times (-2) = -1 \).

This unique relationship helps in quickly identifying perpendicular lines just by looking at their slopes.
Parallel Lines
Parallel lines have a unique property: they never meet, no matter how far they are extended. The reason they do not intersect is that they have the same slope.

When two lines are parallel, the slope \(m\) of both lines is equal. If you have two linear equations in the form \(y = m_1 x + b_1\) and \(y = m_2 x + b_2\), and \( m_1 = m_2\), then the lines are parallel.

For instance, if you have lines with equations \(y = 3x + 2\) and \(y = 3x - 4\), both lines have the same slope of \(3\), which means they are parallel.

Remember that the y-intercepts can be different for parallel lines. It is the equal slopes that define their parallel nature.

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