- Identify the slope of line \(c\) as \(m_c = -6\).
- Determine the slope of the perpendicular line \(d\) using the relationship \(m_d = -1/m_c\), resulting in \(m_d = 1/6\).
- Apply the point-slope formula \(y - y_1 = m(x - x_1)\) with the point \((3, -3)\) and slope \(1/6\).
- Simplify the resulting equation to the slope-intercept form to find the final equation: \(y = \frac{1}{6}x - \frac{7}{2}\).
- Final answer: \(\boxed{y = \frac{1}{6}x - \frac{7}{2}}\).
Explanation
- Analyze Line c
To find the equation of line \(d\), we first examine the given information about line \(c\). The equation of line \(c\) is given as \(y = -6x - 3\). This is in slope-intercept form, \(y = mx + b\), where \(m\) is the slope. Therefore, the slope of line \(c\), which we will denote as \(m_c\), is \(-6\).
- Determine the Slope of Line d
Line \(d\) is perpendicular to line \(c\). For two lines with slopes \(m_1\) and \(m_2\) to be perpendicular, the product of their slopes must be \(-1\), expressed as \(m_1 \cdot m_2 = -1\). Given \(m_c = -6\), we can find the slope of line \(d\) (\(m_d\)) by solving \(m_d \cdot (-6) = -1\). This yields \(m_d = \frac{-1}{-6} = \frac{1}{6}\).
- Use the Point-Slope Form
We know that line \(d\) passes through the point \((3, -3)\) and has a slope of \(m_d = \frac{1}{6}\). We can use the point-slope form, \(y - y_1 = m(x - x_1)\), to find the equation of the line. Substituting the given point \((x_1, y_1) = (3, -3)\) and the slope \(m = \frac{1}{6}\), we get:
\[y - (-3) = \frac{1}{6}(x - 3)\]
\[y + 3 = \frac{1}{6}x - \frac{3}{6}\]
Simplifying the fraction \(\frac{3}{6}\) to \(\frac{1}{2}\), we have:
\[y + 3 = \frac{1}{6}x - \frac{1}{2}\]
- Final Equation Calculation
Finally, we convert the equation to slope-intercept form (\(y = mx + b\)) by isolating \(y\). Subtracting \(3\) from both sides of the equation:
\[y = \frac{1}{6}x - \frac{1}{2} - 3\]
Converting \(3\) to a fraction with a denominator of \(2\), we get \(3 = \frac{6}{2}\). Thus:
\[y = \frac{1}{6}x - \frac{1}{2} - \frac{6}{2}\]
\[y = \frac{1}{6}x - \frac{7}{2}\]
So, the equation of line \(d\) is \(\boxed{y = \frac{1}{6}x - \frac{7}{2}}\).
Examples
This concept of perpendicular lines is essential in fields like urban planning and civil engineering. For example, when designing a road grid or an intersection, engineers need to ensure that streets meet at right angles (90 degrees). By understanding how to calculate slopes and determine perpendicular equations, they can accurately plot the layout of streets and building foundations on a coordinate plane, ensuring structural integrity and geometric precision in their designs.