Determine the equation of a line perpendicular to 4x - 3y - 2 = 0 with the same y-intercept as the line defined by 3x + 4y = - 12
+134 This problem has a lot of steps, so I'll go through each of them one by one. This answer may be long, so if you just want the answer, look to the end, but read the whole thing if you want to understand how I derived it.
So, we want to find the equation of a line. The easiest form to write the equation for this line would be slope-intercept form (y=mx+b) because we are supposed to find the slope of the line perpendicular to another line and the y-intercept of another line.
First, let us find the y-intercept.
We have the line:
3x + 4y = -12
So let's put this into slope intercept form.
3x + 4y = -12
(Subtract 3x)
4y = -3x - 12
(Divide by 4)
y = -3/4x - 3
Now, we know that the y-intercept of the line we need to find the equation of is -3. This is because the b in y = mx + b is the y-intercept.
The next step is to find the slope. To find this, we need to do the same thing as before. Let us put the line 4x - 3y - 2 = 0 into slope-intercept form.
4x - 3y - 2 = 0
(Add 2)
4x - 3y = 2
(Subtract 4x)
-3y = -4x + 2
(Divide by -3)
y = 4/3x + -2/3
This is our new line in slope-intercept form. To now figure out the slope, we need to find the m in y = mx + b and we know this is 4/3.
Now, we have both our slope (4/3) and our y-intercept (-3). From this, we can make the equation of a line in slope-intercept form, but first I must inform you that because we are finding a line perpendicular to the line with that slope, we need the negative reciprocal of the slope (just multiply the slope by -1 and switch the numerator and denominator) equal to -3/4.
The equation of the line perpendicular to 4x - 3y - 2 = 0 with the same y-intercept as the line defined by 3x + 4y = -12 is:
y = mx + b
y = -3/4x - 3
Hope this helps :)
