Lines A and B are perpendicular and have the same x-intercept. The equation of line A is x+2y-4=0. Determine the y-intercept of line B.
Well, I'll have to put on my clown shoes for this one!
Since lines A and B are perpendicular, we know that the slope of line B is the negative reciprocal of the slope of line A.
The equation of line A is x + 2y - 4 = 0. Let's rearrange it to solve for y:
2y = -x + 4
y = (-1/2)x + 2.
So the slope of line A is -1/2.
Since line B is perpendicular to A, the slope of B will be the negative reciprocal of -1/2, which is 2.
Now, we know that lines A and B have the same x-intercept. In other words, when x = 0, both lines pass through the same point.
Substituting x = 0 into the equation of line A, we get y = 2. So, the y-intercept of line A is 2.
Since lines A and B have the same x-intercept, and we already found the y-intercept of line A, we can conclude that the y-intercept of line B is also 2.
So, the y-intercept of line B is 2. Keep those clown shoes on and keep those math questions coming!
To find the y-intercept of line B, we need to determine the equation of line B given that it is perpendicular to line A and has the same x-intercept.
Given line A: x + 2y - 4 = 0, let's rearrange it to the slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
x + 2y - 4 = 0
2y = -x + 4
y = (-1/2)x + 2
We know that line B is perpendicular to line A, which means its slope is the negative reciprocal of the slope of line A. The slope of line A is -1/2, so the slope of line B will be 2.
Now, we know that both lines A and B have the same x-intercept. The x-intercept occurs when y = 0. So, we can substitute y = 0 into the equation of line A to find the x-intercept:
0 = (-1/2)x + 2
(-1/2)x = -2
x = 4
Since both lines A and B have the same x-intercept, line B passes through the point (4, 0).
Now, let's use the point-slope form of a line (y - y1 = m(x - x1)) to find the equation of line B using the known point (4, 0) and the slope of 2:
y - 0 = 2(x - 4)
y = 2x - 8
The equation of line B is y = 2x - 8.
To determine the y-intercept, we can compare the equation of line B (y = 2x - 8) with the slope-intercept form (y = mx + b). We can see that the y-intercept of line B is -8.
Therefore, the y-intercept of line B is -8.
To find the y-intercept of line B, we first need to determine the equation of line B. Since lines A and B are perpendicular, their slopes are negative reciprocals of each other.
Let's start by finding the slope of line A. The given equation of line A is x + 2y - 4 = 0. We can rearrange this equation to the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
x + 2y - 4 = 0
2y = -x + 4
y = -1/2x + 2
From the equation above, we can see that the slope of line A is -1/2.
Since line B is perpendicular to line A, the slope of line B will be the negative reciprocal of -1/2, which is 2/1 or simply 2.
Now we have the slope, m = 2, of line B, and we know that line B has the same x-intercept as line A. The x-intercept of line A can be found by setting y = 0 in the equation y = -1/2x + 2.
0 = -1/2x + 2
1/2x = 2
x = 4
So, the x-intercept of line A is 4.
Now, we can use the point-slope form of a line to determine the equation of line B. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Since line B has the same x-intercept as line A, x1 = 4. Also, since we want to find the y-intercept of line B, we know that when x = 0, y will give us the y-intercept.
Using the point-slope form of line B, we have:
y - y1 = m(x - x1)
y - y1 = 2(x - 4)
y - y1 = 2x - 8
Since x1 = 4, the equation simplifies to:
y - y1 = 2x - 8
y - y1 = 2x - 8
Now, we can substitute x = 0 into this equation to find the y-intercept:
y - y1 = 2(0) - 8
y - y1 = -8
y = -8 + y1
So, the y-intercept of line B is -8 + y1.
Therefore, the y-intercept of line B will depend on the value of y1, which is specific to line B and has not been given in the problem statement.
slope of given line = -1/2
so slope of perpendicular line = +2
x-intercept of given line, let y = 0
x-intercept is (4,0)
so new line: slope = 2, point (4,0) on it
y = 0 = 2(x-4)
y = 2x - 8
y-intercept of new line is -8 or (0, -8)