find the equation, in function form, of athe line that passes through the points(1/2,-3) and (7/6,5/9)
find slope
slope = (5/9 + 3)/(7/6 - 1/2)
= (32/9) / (2/3)
= (32/9)(3/2) = 16/3
so y =(16/3)x + b
sub in the "easier" of the two points, (1/2 , -3)
-3 = (16/3)(1/2) + b
-3 = 8/3 + b
b = -17/3
y = f(x) = (-16/3)x - 17/3
To find the equation of a line that passes through two points, we can use the point-slope form of the equation of a line:
y - y1 = m(x - x1)
where (x1, y1) are the coordinates of one of the points on the line, and m is the slope of the line.
Step 1: Find the slope (m)
The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Given the points (1/2, -3) and (7/6, 5/9), we can substitute the values into the slope formula:
m = (5/9 - (-3)) / (7/6 - 1/2)
m = (5/9 + 27/9) / (7/6 - 3/6)
m = (32/9) / (4/6)
m = (32/9) * (6/4)
m = (32/9) * (3/2)
m = 96/18
m = 16/3
So, the slope of the line passing through the two points is 16/3.
Step 2: Choose one of the points
Let's choose the point (1/2, -3) as our starting point (x1, y1).
Step 3: Write the equation using the point-slope form
Using the point-slope form (y - y1 = m(x - x1)), we substitute the values:
y - (-3) = (16/3)(x - 1/2)
y + 3 = (16/3)(x - 1/2)
y + 3 = (16/3)x - 8/3
Step 4: Convert the equation to function form
To convert the equation to function form (y = f(x)), we isolate y:
y = (16/3)x - 8/3 - 3
y = (16/3)x - 8/3 - 9/3
y = (16/3)x - 17/3
Therefore, the equation, in function form, of the line passing through the points (1/2, -3) and (7/6, 5/9) is:
y = (16/3)x - 17/3.