Question

which of the following equations represents a line parallel to $y - 6 = 12(x + 2)$ that passes through $(2, 3)$?
$y - 3 = -\frac{1}{12}(x - 2)$
$y - 6 = -\frac{1}{12}(x + 2)$
$y - 3 = 12(x - 2)$
$y - 6 = 12(x - 2)$

Explanation:

Step1: Recall slope of parallel lines

Parallel lines have equal slopes. The given line is in point - slope form \(y - y_1=m(x - x_1)\), where \(m\) is the slope. For the line \(y - 6=12(x + 2)\), the slope \(m = 12\). So the line we are looking for must also have a slope of 12.

Step2: Use point - slope form for the new line

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope. We know the line passes through \((2,3)\) and has a slope \(m = 12\). Substituting \(x_1 = 2\), \(y_1=3\) and \(m = 12\) into the point - slope form, we get \(y - 3=12(x - 2)\).

Answer:

\(y - 3 = 12(x - 2)\) (the third option)