- Identify the slope of line \(p\) from its point-slope equation as \(m_p = \frac{1}{4}\).
- Calculate the slope of the perpendicular line \(q\) using the negative reciprocal relationship: \(m_q = -4\).
- Apply the point-slope formula with the point \((3, -6)\) to derive the equation \(y + 6 = -4(x - 3)\).
- Simplify the equation to reach the final slope-intercept form: \(\boxed{y = -4x + 6}\).
Explanation
- Analyze the given information and identify the slope of line \(p\).
The given line \(p\) has the equation \(y - 3 = \frac{1}{4}(x + 1)\). This equation is in point-slope form, \(y - y_1 = m(x - x_1)\), where the slope \(m\) is the coefficient of the parenthesis. Comparing the two, we identify the slope of line \(p\) as \(m_p = \frac{1}{4}\). We are looking for the equation of a line \(q\) that is perpendicular to line \(p\) and passes through the point \((3, -6)\).
- Determine the slope of line \(q\).
Two lines are perpendicular if the product of their slopes is \(-1\). Let \(m_q\) be the slope of line \(q\). Thus, \(m_p \times m_q = -1\). Substituting the known slope of line \(p\):
\[\frac{1}{4} \times m_q = -1\]
Solving for \(m_q\):
\[m_q = -1 \times 4 = -4\]
The slope of line \(q\) is \(-4\).
- Find the equation of line \(q\) in slope-intercept form.
Now we have the slope \(m_q = -4\) and a point \((3, -6)\) that lies on line \(q\). Using the point-slope formula \(y - y_1 = m(x - x_1)\):
\[y - (-6) = -4(x - 3)\]
Simplify the equation:
\[y + 6 = -4x + 12\]
To write the equation in slope-intercept form (\(y = mx + b\)), isolate \(y\):
\[y = -4x + 12 - 6\]
\[y = -4x + 6\]
Thus, the equation of line \(q\) is \(y = -4x + 6\).
Examples
In urban planning, streets are often designed to intersect at right angles to create orderly city blocks. If one street follows the path of the line \(y - 3 = \frac{1}{4}(x + 1)\), then a perpendicular street (a cross-street) passing through a specific intersection point like \((3, -6)\) must follow a predictable path determined by the perpendicular slope \(-4\). This mathematical principle helps engineers ensure that street layouts are geometrically precise and functionally efficient.