Answer:
Equation (in slope-intercept form): y = 3/4x + 9
Equation (in standard form): -3x + 4y = 36
You can use either equation as both are correct with the only difference being the form.
Explanation:
Relationship between the slopes of perpendicular lines:
- The slopes of perpendicular lines are negative reciprocals of each other.
We can represent this with the equation m2 = -1/m1, where:
- m1 is the slope of the line we're given,
- and m2 is the slope of the line we're trying to find.
- 4x + 3y = -24 is in the standard form of a line, but we can can identify its slope by converting it slope-intercept form.
Converting 4x + 3y = -24 to slope-intercept form and identifying its slope (i.e., m1):
The general equation of the slope-intercept form is given by:
y = mx + b, where:
- m is the slope,
- and b is the y-intercept.
Thus, we can can convert 4x + 3y = -24 to slope-intercept form by isolating y:
(4x + 3y = -24) - 4x
(3y = -4x - 24) / 3
y = -4/3x - 8
Thus, the slope of 4x + 3y = -24 is -4/3.
Finding the slope of the other line (i.e., m2):
Now, we can find the slope of the other line by substituting -4/3 for m1 in the perpendicular slope equation:
m2 = -1 / (-4/3)
m2 = -1 * (-3/4)
m2 = 3/4
Thus, the slope of the other line is 3/4.
Finding the y-intercept of the other line and writing the equation of the line:
Now, we can find the y-intercept (b) of the other line by substituting (-8, 3) for (x, y) and 3/4 for m in the slope-intercept form:
3 = 3/4(-8) + b
3 = -24/4 + b
(3 = -6 + b) + 6
9 = b
Thus, the y-intercept of the other line is 9.
Therefore, y = 3/4x + 9 is the equation of a line perpendicular to 4x + 3y = -24 and passing through the point (-8, 3).
----------------------------------------------------------------------------------------------------------Converting y = 3/4x + 9 to standard form:
- Since the other line is given in standard form, we can convert y = 3/4x + 9 to standard form for the sake of continuity.
First, let's clear the fraction by multiplying the entire equation by 4:
4(y = 3/4x + 9)
4y = 3x + 36
Now, we need to isolate 36 on the right-hand side:
(4y = 3x + 36) - 3x
-3x + 4y = 36
Thus, -3x + 4y = 36 is also the equation of a line perpendicular to 4x + 3y = -24 and passing through the point (-8, 3).