Find the equation, in standard form, with all integer coefficients, of the line perpendicular to x + 3y = 12 and passing through (9, -5).

Their slopes must negative reciprocals of each other. So the new equation must be

3x - y = c

but (9,-5) lies on it, so
3(9) - (-5) = c
c = 32
and your equation is
3x - y = 32

To find the equation of a line perpendicular to the given line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.

The given equation is in the form Ax + By = C, where A, B, and C are constants. We can rewrite the given equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

So, let's rewrite the given equation in slope-intercept form:
x + 3y = 12
3y = -x + 12
y = (-1/3)x + 4

From this equation, we can see that the slope of the given line is -1/3.

To find the slope of the line perpendicular to this, we need to take the negative reciprocal of -1/3. The negative reciprocal is obtained by flipping the fraction and changing the sign. The negative reciprocal of -1/3 is 3/1 or simply 3.

Now, we have the slope of the line that is perpendicular to the given line, which is 3. We also know that this line passes through the point (9, -5).

Using the point-slope form of a linear equation (y - y1 = m(x - x1)), where (x1, y1) is a specific point on the line, and m is the slope, we can substitute the values into the equation to find the equation of the line:

y - (-5) = 3(x - 9)
y + 5 = 3x - 27
y = 3x - 32

Therefore, the equation of the line perpendicular to x + 3y = 12 and passing through (9, -5) is y = 3x - 32, which is in standard form with all integer coefficients.