Determine if the lines represented by the equations x - 2y = 5 and 2x + y = -2 are parallel, perpendicular, or neither.

Understand the Problem

The question asks to determine the relationship between the two given equations, specifically whether they are parallel, perpendicular, or neither based on their slopes.

Answer

The lines are perpendicular.

Steps to Solve

Rewrite the first equation in slope-intercept form (y = mx + b)

The first equation is given as $x - 2y = 5$. To convert it to slope-intercept form, isolate $y$:

$$ -2y = -x + 5 $$

Dividing by -2 gives:

$$ y = \frac{1}{2}x - \frac{5}{2} $$

Here, the slope ($m_1$) is $\frac{1}{2}$.

Rewrite the second equation in slope-intercept form

The second equation is $2x + y = -2$. To convert it to slope-intercept form, isolate $y$:

$$ y = -2x - 2 $$

Here, the slope ($m_2$) is $-2$.

Determine the relationship between the slopes

Check the condition for parallel or perpendicular lines:

Two lines are parallel if their slopes are equal: $m_1 = m_2$.
Two lines are perpendicular if the product of their slopes is $-1$: $m_1 \cdot m_2 = -1$.

Calculating the product of the slopes:

$$ m_1 \cdot m_2 = \left(\frac{1}{2}\right) \cdot (-2) = -1 $$

Thus, the lines are perpendicular.

The lines are perpendicular.

More Information

In geometry, perpendicular lines intersect at right angles (90 degrees). The relationship between the slopes is crucial: when the product of two slopes equals -1, it indicates that the lines are perpendicular.

Tips

Confusing parallel with perpendicular: It's a common mistake to misidentify the relationship; remember that parallel lines have equal slopes, while perpendicular lines have slopes that multiply to -1.
Incorrect rearrangement of equations: Ensure you are correctly isolating $y$ in slope-intercept form—errors in this step can lead to wrong slope calculations.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!