In the xy-plane, line m passes through the origin and is perpendicular to the line 8x-3y=n, where n is a constant. If the two lines intersect at the point (r, r-2), what is the value of r?
Line m has slope -3/8. So, using the point-slope form,
y-(r-2) = -3/8 (x-r)
or,
3x+8y = 11r-16
Now solve that along with 8x-3y=n and set the intersection to be (r,r-2). Then solve for r in terms of n.
8
To find the value of r, we need to find the equation of line m and then find the point of intersection with the line given by 8x - 3y = n.
Step 1: Find the slope of line m.
Since line m is perpendicular to the line 8x - 3y = n, the slope of line m will be the negative reciprocal of the slope of 8x - 3y = n.
The slope-intercept form of the equation 8x - 3y = n is: y = (8/3)x - (n/3).
So, the slope of the line is 8/3.
The negative reciprocal of 8/3 is -3/8.
Step 2: Find the equation of line m.
Since line m passes through the origin (0, 0) and has a slope of -3/8, we can write the equation of line m in slope-intercept form: y = mx.
Substituting the slope, we get y = (-3/8)x.
Step 3: Find the point of intersection.
We know that the point of intersection is (r, r-2). We can substitute these values into the equations of line m and 8x - 3y = n to find the value of r.
Substituting into the equation of line m, we get: r-2 = (-3/8)r.
Multiplying through by 8 to eliminate the fraction, we get: 8(r-2) = -3r.
Expanding, we have: 8r - 16 = -3r.
Combining like terms, we get: 11r = 16.
Dividing by 11, we find: r = 16/11.
Therefore, the value of r is 16/11.
To find the value of r, we need to understand the properties of perpendicular lines and solve the given equation system.
Let's first analyze the line 8x - 3y = n. We can rearrange it into the slope-intercept form (y = mx + b) by isolating y:
8x - 3y = n
-3y = -8x + n
y = (8/3)x - (n/3)
We know that line m is perpendicular to this line and passes through the origin (0, 0). Since two lines are perpendicular, their slopes are negative reciprocals of each other.
The slope of the line 8x - 3y = n can be determined by comparing it with the standard equation y = mx + b, where m represents the slope. From the equation, we can deduce that the slope is 8/3.
To find the slope of line m, we take the negative reciprocal of 8/3, which gives us -3/8.
Now, we have the slope of line m, which is -3/8, and it passes through the origin (0, 0).
Using the point-slope form of a line (y - y1 = m(x - x1)) and substituting the values we have:
y - 0 = (-3/8)(x - 0)
y = (-3/8)x
Next, we need to find the point of intersection between line m and the line 8x - 3y = n. To do this, we'll set the two equations equal to each other and solve for x.
(-3/8)x = (8/3)x - (n/3)
Multiply both sides by 24 to eliminate fractions:
-9x = 64x - 8n
Combining like terms:
64x + 9x = 8n
73x = 8n
x = 8n/73
Now, we can substitute this value of x back into one of the equations to find the value of n. Let's use the equation 8x - 3y = n:
8(8n/73) - 3y = n
64n/73 - 3y = n
64n - 73(3y) = 73n
64n - 219y = 73n
219y = 9n
y = 9n/219
So, the point of intersection is (x, y) = (8n/73, 9n/219). Given that this point is (r, r-2), we can set the x and y coordinates equal to r and r-2, respectively:
r = 8n/73
r-2 = 9n/219
To find the value of r, we can solve these two equations simultaneously. We'll start by isolating n in the first equation:
73r = 8n
n = 73r/8
Now, substitute this expression for n into the second equation:
r-2 = 9(73r/8)/219
219r - 2*219 = 9 * 73r
219r - 438 = 657r
438 = 438r
r = 1
Therefore, the value of r is 1.