Answer:
(A) The equation of the line perpendicular to the line 6x - 4y = -8 and passes through (4, -6) is y = -2/3x - 10/3
(B) The equation of the line that is parallel to the line 6x - 4y = -8 and pases through (4, -6) is y = 3/2x - 12
Explanation:
(A)
- The slopes of perpendicular lines are negative reciprocals of each other, as shown by the formula m2 = -1 / m1, where
- m2 is the slope of the line we don't know,
- and m1 is the slope of the line we're given.
Currently, 6x - 4y = -8 is in standard form, but we can convert it to slope-intercept form (y = mx + b with the slope being m) by isolating y:
Step 1: Subtract 6x from both sides:
(6x - 4y = -8) - 6x
-4y = -6x - 8
Step 2: Divide both sides by -4 to isolate y and find the slope-intercept form:
(-4y = -6x - 8) / -4
y = (-6x) / -4 + (-8) / -4
y = 3/2x + 2
Thus, the slope of the line we're given (aka m1 in the perpendicular slope formula) is 3/2.
Step 3: Now we can plug in 3/2 for m1 in the perpendicular slope formula to find m2, the slope of the other line:
m2 = -1 / (3/2)
m2 = -1 * 2/3
m2 = -2/3
Thus, the slope of the other line is -2/3
Step 4: We can keep using the slope-intercept form to find b, the y-intercept of the line. To do this, we must plug in (4, -6) for x and y and -2/3 for m, allowing us to solve for b:
y = mx + b
-6 = 4(-2/3) + b
-6 = -8/3 + b
-10/3 = b
Thus, the equation of the line perpendicular to the line 6x - 4y = -8 and passes through (4, -6) is y = -2/3x - 10/3
(B)
- The slopes of parallel lines are equal to each other, as shown by the formula m2 = m1, where
- m1 is the slope we're given,
- and m2 is the slope of the other line
Step 1: We already know that m1 is 3/2 so m2 is also 3/2. Thus, the slope of the other line is 3/2
Step 2: We can use the slope-intercept form to find b, the y-intercept of the other line. To do this, we must plug in (4, -6) for x and y and 3/2 for m, allowing us to solve for b:
y = mx + b
-6 = 3/2(4) + b
-6 = 12/2 + b
-6 = 6 + b
-12 = b
Thus, the equation of the line that is parallel to the line 6x - 4y = -8 and passes through (4, -6) is y = 3/2x - 12