What is the equation of a line that is perpendicular to the line represented by the equation y = 12x + 5 and passes through the point (6, -4)?

To find the equation of a line that is perpendicular to the given line and passes through the specified point, follow these steps:

Step 1: Determine the slope of the original line

The given line’s equation is in the slope-intercept form: y = mx + b, where m is the slope. Here, the slope of the line is:

m = 12

Step 2: Calculate the slope of the perpendicular line

Lines that are perpendicular to each other have slopes that are negative reciprocals. Therefore, to find the slope of the line that is perpendicular to the given line:

If m = 12, then the slope m_{perpendicular} = -1/m = -1/12.

Step 3: Use the point-slope form of the equation

Now that we have the slope of the perpendicular line, we can use the point-slope form of the equation of a line, which is:

y – y₁ = m(x – x₁)

Here, (x₁, y₁) is the point through which our new line passes, which is (6, -4), and m is the slope we calculated:

y – (-4) = -1/12(x – 6)

Step 4: Simplify the equation

Now we will simplify the equation:

Start with:

y + 4 = -1/12(x – 6)

Distribute -1/12:

y + 4 = -1/12x + 1/2

Subtract 4 from both sides:

y = -1/12x + 1/2 – 4

Convert 4 to have a denominator of 2:

y = -1/12x + 1/2 – 8/2

Simplify it:

y = -1/12x – 7/2

Final Result

The equation of the line that is perpendicular to the line represented by y = 12x + 5 and passes through the point (6, -4) is: