Let's begin by finding the slope of the given line by transforming its equation into slope-intercept form,
(y = mx + b), where m is the slope and b is the y-intercept:
9x - 9y = 4 (equation of given line)
-9y = -9x + 4
y = x -(4/9)
The slope (m) of this line is 1 (coefficient of x). The slope of the perpendicular line will be the negative reciprocal of the slope of the given line.
Therefore, the slope of the perpendicular line will be the negative reciprocal of 1 (-1) and, so the slope-intercept form of its equation will be:
y = -x + b
We are told that the y-intercept (b) of this line will be the same value as the y-intercept (b) of the line with equation (6x + 4y = 6). So we'll need to transform the equation of this line into slope-intercept form to see what is its y-intercept (b):
6x + 4y = 6
4y = -6x + 6
y = -(3/2)x + 3/2 (b = 3/2)
So we will plug in value of 3/2 for b into slope-intercept form of equation for our perpendicular line and we're done:
y = -x + b
y = -x + 3/2 (our answer)
This problem required a two-prong approach:
1. Find the slope of the given line and take its negative reciprocal for slope of our perpendicular line.
2. Find the y-intercept of the another line and plug it into slope-intercept form of equation for our perpendicular line.
Thanks for submitting this problem and glad to help.
God bless, Jordan.