Find an equation for the line with y-intercept 3 that is perpendicular to the line 3y=2x-4.
A. 2y=6-3x
B. 2y=3x+6
C. 3y=9-2x
D. 3y=2x+9
y = -3x/2 + 3
Multiply both sides by -2:
-2y = 3x - 6
2y = -3x + 6
2y = 6 - 3x
Answer = A.
Well, finding a line perpendicular to another line is like finding a friend who completely contradicts everything you say. It's a real challenge!
To solve this problem, we need to find the slope of the given line. Since the equation is in the form of 3y=2x-4, we can divide everything by 3 to get y=(2/3)x-4/3. That means the slope of the given line is 2/3.
Now, if we want a line to be perpendicular to this line, we need to find the negative reciprocal of the slope. Remember, in the world of lines, like in friendship, opposites attract! So, the negative reciprocal of 2/3 is -3/2.
Since we also know the line has a y-intercept of 3, we can use the point-slope form of a line: y - y1 = m(x - x1). Plugging in the y-intercept value, we get y - 3 = -3/2(x - 0). Simplifying, we get y - 3 = -3/2x.
Multiplying through by 2, we get 2y - 6 = -3x. Rearranging the equation, we get the line in the form 3x + 2y - 6 = 0. And if you're a fan of multiple choice, that matches option B: 2y = 3x + 6.
So, the equation for the line with a y-intercept of 3 that's perpendicular to 3y = 2x - 4 is B. 2y = 3x + 6. Keep calm and calculate on!
To find the equation of a line that is perpendicular to another line, we need to find the negative reciprocal of the slope of the given line.
The given line is 3y = 2x - 4. To find its slope, we can rewrite it in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
So, dividing the equation by 3:
y = (2/3)x - 4/3
The slope of the given line is 2/3.
The negative reciprocal of 2/3 is -3/2.
Since we want a line with a y-intercept of 3, we can use the point-slope form of the line, which is:
y - y1 = m(x - x1)
Substituting the values, we get:
y - 3 = (-3/2)(x - 0)
Simplifying:
y - 3 = (-3/2)x
Multiplying by -2 to eliminate the fraction:
2(y - 3) = -3x
Expanding:
2y - 6 = -3x
Rearranging to the standard form:
3x + 2y = 6
Therefore, the equation of the line with a y-intercept of 3 that is perpendicular to the line 3y = 2x - 4 is 3x + 2y = 6.
So, the correct answer is D. 3y = 2x + 9.
To find an equation for a line that is perpendicular to another line, we need to consider the slopes of the two lines. Two lines are perpendicular if and only if the product of their slopes is -1.
Given the line 3y = 2x - 4, we can rewrite it in slope-intercept form (y = mx + b) by dividing both sides of the equation by 3:
y = (2/3)x - 4/3
The slope of this line is 2/3, so the slope of a line perpendicular to it would be the negative reciprocal, which is -3/2.
Now, we know that our line has a y-intercept of 3. The y-intercept is the point where the line intersects the y-axis, so it is the value of y when x = 0. Since the line passes through (0,3), we can plug this point into the slope-intercept form (y = mx + b) to find the equation:
3 = (-3/2)(0) + b
3 = b
Therefore, the equation for the line with a y-intercept of 3 that is perpendicular to the line 3y = 2x - 4 is y = (-3/2)x + 3.
Comparing this equation to the answer choices:
A. 2y = 6 - 3x (Not the correct equation)
B. 2y = 3x + 6 (Not the correct equation)
C. 3y = 9 - 2x (Not the correct equation)
D. 3y = 2x + 9 (Correct equation)
The correct equation is D. 3y = 2x + 9.