Question

write the equation of a line that is perpendicular to the given line and that passes through the given point.

1. ( x + 12y = 29 ); ( (5, 9) )

Explanation:

Step1: Find slope of given line

Rewrite \(x + 12y = 29\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope).
Subtract \(x\) from both sides: \(12y=-x + 29\).
Divide by 12: \(y=-\frac{1}{12}x+\frac{29}{12}\).
The slope of the given line \(m_1 =-\frac{1}{12}\).

Step2: Find slope of perpendicular line

If two lines are perpendicular, the product of their slopes \(m_1\times m_2=- 1\).
Let \(m_2\) be the slope of the perpendicular line. Then \(-\frac{1}{12}\times m_2=-1\).
Solve for \(m_2\): \(m_2 = 12\) (multiply both sides by - 12).

Step3: Use point - slope form to find equation

Point - slope form is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(5,9)\) and \(m = 12\).
Substitute values: \(y - 9=12(x - 5)\).

Step4: Simplify to slope - intercept form (optional, but common)

Expand: \(y - 9=12x-60\).
Add 9 to both sides: \(y=12x-60 + 9\).
Simplify: \(y=12x-51\). Or in standard form \(12x-y=51\).

Answer:

The equation of the line is \(y = 12x-51\) (or \(12x - y=51\))