please help me. find the equation of a line through the points (2,8) and (1,5) in the form of x/a + y/b =1 where a and b are y and x intercepts
plug in your two points:
2/a + 8/b = 1
1/a + 5/b = 1
multiply the 2nd by 2 and you have
2/a + 8/b = 1
2/a + 10/b = 2
now subtract, and you have
2/b = 1
b = 2
use that to find that a=-2/3
x/(-2/3) + y/2 = 1
check the intercepts and you see that they are (-2/3,0) and (0,2)
Now you know why that's called the intercept form of the line.
The two-point form of a straigth line:
y - y1 = [ ( y2 - y1 ) / ( x2 - x1 ) ] · ( x - x1 )
In this case :
x1 = 2 , y1 = 8
x2 = 1 , y2 = 5
So:
y - y1 = [ ( y2 - y1 ) / ( x2 - x1 ) ] · ( x - x1 )
y - 8 = [ ( 5 - 8 ) / ( 1 - 2 ) ] · ( x - 2 )
y - 8 = [ ( - 3 ) / ( - 1 ) ] · ( x - 2 )
y - 8 = 3 · ( x - 2 )
y - 8 = 3 · x - 3 · 2
y - 8 = 3 x - 6
Add 8 to both sides
y - 8 + 8 = 3 x - 6 + 8
y = 3 x + 2
y = m · x + b
This is the "slope-intercept" form.
m is the slope
b is the y - intercept
In this case:
y - intercept = b = 2
Now you must find x - intercept
x - intercept is point where y = 0
y = 3 x + 2 = 0
3 x + 2 = 0
Subtract 2 to both sides
3 x + 2 - 2 = 0 - 2
3 x = - 2
Divide both sides by 3
x = - 2 / 3
x / a + y / b = 1
is the intercept form of the equation of a straigth line
a is the x - intercept
b is the y - intercept
In this case:
x - intercept = a = - 2 / 3
y - intercept = b = 2
x / a + y / b = 1
x / ( - 2 / 3 ) + y / 2 = 1
( x / 1 ) / ( - 2 / 3 ) + y / 2 = 1
( x · 3 ) / ( - 2 · 1 ) + y / 2 = 1
3 x / - 2 + y / 2 = 1
- 3 x / 2 + y / 2 = 1
Sure! To find the equation of a line through two points in the form x/a + y/b = 1, where a and b are the x and y-intercepts, we need to follow a few steps:
Step 1: Find the slope (m) of the line using the given points (2,8) and (1,5).
The slope formula is given by: m = (y2 - y1) / (x2 - x1)
Using the coordinates (2,8) and (1,5), the slope can be calculated as follows:
m = (5 - 8) / (1 - 2) = (-3) / (-1) = 3
Step 2: Use the slope-intercept form, y = mx + b, to find the y-intercept (b).
We can choose either of the given points. Let's use (2,8):
8 = 3(2) + b
8 = 6 + b
b = 8 - 6
b = 2
So, the y-intercept (b) is 2.
Step 3: Find the x-intercept by setting y = 0. We will solve for x.
x/a + y/b = 1
x/a + 0/b = 1
x/a = 1
x = a
Therefore, the x-intercept is a.
Now that we have the slope (m = 3), the y-intercept (b = 2), and the x-intercept (a), we can plug these values into the equation x/a + y/b = 1.
The equation of the line through the points (2,8) and (1,5) in the form x/a + y/b = 1 is:
x/a + y/2 = 1
Please note that the coefficients a and b may have different values depending on the specific line equation found.