DIRECTIONS ARE.
WRITE THE EQUATION OF THE LINE L satisfying the given geometric conditions.
Problem is:
L has y-intercept (0,-3) and is parallel to the line with equation
y=(2/3)x +1.
So, this is what i did:
Y=(2/3)x+b....it passes through (0,-3)
-3=b
so, the equation is:
y=(2/3)x-3
then problem number 2:
L has y-intercept (0,2) and is perpendicular to the line with equation 2x-3y=6.
so this is whati did:
3x+2y=K..it passes through (0,2)
2(2)=k
4=k
so the equation would be:
3x+2y=4
first is correct.
second correct.
thank you for helping me out.
You did a great job in solving both problems! Let me explain the steps to find the equations of the lines in more detail.
Problem 1:
To find the equation of a line parallel to a given line, we need to make use of the fact that parallel lines have the same slope. The given line has the equation y = (2/3)x + 1, which means its slope is 2/3.
Now, let's find the equation of the parallel line with a y-intercept at (0, -3). We start with the general equation y = mx + b, where m represents the slope of the line and b represents the y-intercept.
Since the line is parallel to the given line, we can use the same slope, m = 2/3. Substituting the values, we have y = (2/3)x + b.
To determine the value of b, we use the fact that the line passes through the point (0, -3). Plug in x = 0 and y = -3 into the equation:
-3 = (2/3)(0) + b
-3 = b
Therefore, the equation of the line is y = (2/3)x - 3.
Problem 2:
To find the equation of a line perpendicular to a given line, we need to note that the slopes of perpendicular lines are negative reciprocals of each other. The given line has the equation 2x - 3y = 6, which can be rewritten in slope-intercept form as y = (2/3)x - 2.
The slope of the given line is 2/3. The negative reciprocal of 2/3 is -3/2, which will be the slope of the perpendicular line.
Now, let's find the equation of the perpendicular line with a y-intercept at (0, 2). Again, we start with the general equation y = mx + b.
Substituting the slope, m = -3/2, and the y-intercept, b = 2, we have y = (-3/2)x + 2.
Thus, the equation of the line is 3x + 2y = 4 (multiplying through by 2 to eliminate the fraction).
Therefore, the equations you derived are correct:
1. y = (2/3)x - 3
2. 3x + 2y = 4
I hope this explanation helps you understand how to find equations of lines given specific conditions. If you have any further questions, feel free to ask!