1) L has y intercept (0, -3) and is parrallel to the line with equation 2/3x + 1
2) L has y intercept (0, 2) and is perpendicular to the line with equation 2x -3y = 6
1) you know the slope, and intercept
2) Put the equation in slope intercept form.
To solve these questions, we need to find the equation of the line L in each case.
1) To find the equation of the line L parallel to the line 2/3x + 1, we know that parallel lines have the same slope. Therefore, the slope of line L will also be 2/3.
We also know that line L has a y-intercept of (0, -3). The y-intercept is the point where the line crosses the y-axis, so when x = 0, y = -3.
Using the slope-intercept form of a line (y = mx + b), where m is the slope and b is the y-intercept, the equation of line L can be written as:
y = (2/3)x - 3
2) To find the equation of the line L perpendicular to 2x - 3y = 6, we need to determine the slope first.
The given equation of the line can be rearranged in slope-intercept form as follows:
-3y = -2x + 6
Dividing both sides by -3 gives us:
y = (2/3)x - 2
For a line to be perpendicular to this line, its slope must be the negative reciprocal of (2/3). The negative reciprocal of a number a/b is -b/a. In this case, the negative reciprocal of 2/3 is -3/2.
Since we know that L has a y-intercept of (0, 2), the equation of line L can be written as:
y = (-3/2)x + 2
To summarize:
1) The equation of line L parallel to 2/3x + 1 is y = (2/3)x - 3.
2) The equation of line L perpendicular to 2x - 3y = 6 is y = (-3/2)x + 2.
To find the equation of the line L that is parallel to a given line or perpendicular to another given line, we can follow these steps:
1) For the line that is parallel (Question 1) or perpendicular (Question 2), we need to determine its slope.
For Question 1, the given line has the equation 2/3x + 1. We can identify the slope by comparing it to the slope-intercept form (y = mx + b), where m represents the slope. So in this case, the slope is 2/3.
For Question 2, the given line has the equation 2x - 3y = 6. To determine the slope, we need to rearrange the equation into slope-intercept form. Subtract 2x from both sides of the equation to isolate the -3y term. The equation then becomes -3y = -2x + 6. Divide through by -3 to solve for y: y = (2/3)x - 2/3. Now we can see that the slope is 2/3.
2) Now that we know the slope for both problems, we can use the slope-intercept form (y = mx + b) to find the equation of the lines.
For Question 1, we know that the line L is parallel to the given line and passes through the y-intercept (0, -3). The slope is 2/3. Substitute the values into the equation y = mx + b:
y = (2/3)x + b
Using the y-intercept (-3), we can substitute the x and y coordinates into the equation:
-3 = (2/3)(0) + b
Simplifying, we find that b = -3. Therefore, the equation for line L in Question 1 is y = (2/3)x - 3.
For Question 2, we know that the line L is perpendicular to the given line and passes through the y-intercept (0, 2). The slope is 2/3, but since the line is perpendicular, we need to find the negative reciprocal of the slope. The negative reciprocal of 2/3 is -3/2.
Now, substitute the values into the equation y = mx + b:
y = (-3/2)x + b
Using the y-intercept (2), we can substitute the x and y coordinates into the equation:
2 = (-3/2)(0) + b
Simplifying, we find that b = 2. Therefore, the equation for line L in Question 2 is y = (-3/2)x + 2.