A line whose equation is 7y-3x+30=0 passes through the points A and B . Points A is on -axis while point B is equidistant from X and Y -axis
You might start with the intercept form for the line
If x/a + y/b = 1
then the x- and y- intercepts are at (a,0) and (0,b)
So, your line can be expressed as
3x - 7y = 30
x/10 - y/(30/7) = 1
So now you can easily find A.
B is the intersection of your line and the line y=x, which bisects the angle between the axes.
Not well taught, didn't understand anything
To find the points A and B, we need to solve the equation of the line for the x and y coordinates.
Given the equation of the line: 7y - 3x + 30 = 0
For point A, it lies on the x-axis, which means its y-coordinate is 0. Substituting this into the equation, we have:
7(0) - 3x + 30 = 0
-3x + 30 = 0
-3x = -30
x = 10
Therefore, point A is (10, 0).
For point B, it is equidistant from the x-axis and y-axis. This means the distance of B from the x-axis is the same as its distance from the y-axis. Let's assume the distance is 'd'. Using the distance formula, we have:
Distance from x-axis: |y - 0| = |y|
Distance from y-axis: |x - 0| = |x|
Since B is equidistant from the x and y-axis, we have the following two equations:
|y| = |x + d|
For positive values of x and y, the equation becomes:
y = x + d
Substituting these equations into the line equation, we get:
7(y) - 3(x) + 30 = 0
7(x + d) - 3x + 30 = 0
7x + 7d - 3x + 30 = 0
4x + 7d + 30 = 0
4x = -7d - 30
x = (-7d - 30) / 4
Since x and y are equidistant from the origin, we can equate their absolute values:
|y| = |x + d|
y = x + d
Substituting the value of x, we get:
y = (-7d - 30) / 4 + d
Simplifying the equation, we have:
4y = -7d - 30 + 4d
4y = -3d - 30
4y + 3d = -30
To solve for d, we need another equation. We can use the fact that B lies on the line:
7y - 3x + 30 = 0
Substituting the values of x and y, we have:
7((-7d - 30) / 4) - 3((-7d - 30) / 4) + 30 = 0
Multiplying both sides of the equation by 4 to eliminate the fraction, we get:
7(-7d - 30) - 3(-7d - 30) + 120 = 0
-49d - 210 - 21d - 90 + 120 = 0
-70d - 180 = 0
-70d = 180
d = -180 / -70
d = 36/14
d = 18/7
Substituting this value of d back into the equation we found for y, we have:
y = (-7(18/7) - 30) / 4 + 18/7
y = -18 - 30 / 4 + 18/7
y = (-48 / 4) + 18/7
y = -12 + 18/7
To simplify y, we need to find a common denominator:
y = -12(7/7) + 18/7
y = (-84 / 7) + 18/7
y = (-84 + 18) / 7
y = -66 / 7
Therefore, point B is (-7.714, -9.429).
In summary:
- Point A is (10, 0).
- Point B is approximately (-7.714, -9.429).
To find the points A and B, we need to solve the given equation for the two unknowns, x and y.
1. Point A lies on the y-axis, meaning its x-coordinate is 0. Substitute x = 0 into the equation to find the corresponding y-coordinate:
7y - 3(0) + 30 = 0
7y + 30 = 0
7y = -30
y = -30/7
So, point A is (0, -30/7).
2. Point B is equidistant from the x-axis and y-axis. This means its x-coordinate is equal to its y-coordinate. Let's call this common value "k". So, the coordinates of point B are (k, k).
Substitute these coordinates into the equation to find the value of k:
7(k) - 3(k) + 30 = 0
4k + 30 = 0
4k = -30
k = -30/4
k = -15/2
So, point B is (-15/2, -15/2), or (-7.5, -7.5).
Therefore, the line whose equation is 7y - 3x + 30 = 0 passes through point A (0, -30/7) and point B (-7.5, -7.5).