The line l1 passes through the points (3,-3) and (-5,2). The line is the graph of the equation Ax + By = C, where A, B, and C are integers with greatest common divisor 1, and A is positive. Find A + B+ C.
(2+3)/(-5-3) = (y+3) / (x-3)
5/-8 = (y+3) / (x-3)
5 x -15 = -8 y -24
5 x + 8 y = -9
5+8-9 = 4
To find the equation of the line passing through the points (3, -3) and (-5, 2), we can use the formula for the equation of a line given two points.
First, we need to find the slope of the line. The slope (m) is given by the formula:
m = (y2 - y1)/(x2 - x1)
Let (x1, y1) = (3, -3) and (x2, y2) = (-5, 2).
m = (2 - (-3))/(-5 - 3)
m = 5/(-8)
m = -5/8
Now that we have the slope, we can use the point-slope form of a line to find the equation:
y - y1 = m(x - x1)
Using (x1, y1) = (3, -3):
y - (-3) = (-5/8)(x - 3)
y + 3 = (-5/8)(x - 3)
To get rid of the fraction, let's multiply both sides of the equation by 8:
8(y + 3) = -5(x - 3)
Expanding and simplifying:
8y + 24 = -5x + 15
8y + 5x = -9
Now we have the equation in the form Ax + By = C, where A = 5, B = 8, and C = -9.
Finally, to ensure that A, B, and C have the greatest common divisor of 1 and A is positive, we divide all terms of the equation by -1:
-8y - 5x = 9
Therefore, A = -5, B = -8, and C = 9.
To find A + B + C:
A + B + C = -5 + (-8) + 9
A + B + C = -13 + 9
A + B + C = -4
To find the equation of the line passing through the given points, we can first find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points on the line.
In this case, let's use the points (3, -3) and (-5, 2):
m = (2 - (-3)) / (-5 - 3)
= (2 + 3) / (-5 - 3)
= 5 / (-8)
= -5/8
Now that we have the slope, we can use the point-slope form of the equation of a line:
y - y1 = m(x - x1)
Using the point (3, -3), we get:
y - (-3) = -5/8(x - 3)
y + 3 = -5/8(x - 3)
To convert the equation into standard form (Ax + By = C), we multiply both sides of the equation by 8 to eliminate the fraction:
8(y + 3) = -5(x - 3)
8y + 24 = -5x + 15
5x + 8y = 15 - 24
5x + 8y = -9
Now we have the equation in the required form of Ax + By = C, where A = 5, B = 8, and C = -9.
Finally, we can find A + B + C:
A + B + C = 5 + 8 + (-9) = 4