The line l_1 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.
Find Slope:
m = (2 - -3)/(-5 - 3) = -5/8
Plug one of the points in Point-Slope Equation:
y - y₁ = m (x - x₁)
y - 2 = -5/8 (x + 5)
Rearrange the equation to make it in Standard Form:
y - 2 = -5/8 (x + 5)
y - 2 = -5x/8 - 25/8
8y - 16 = -5x - 25
5x + 8y = -9
Find A + B + C
5 + 8 + (-9) = 4
To find the equation of the line passing through the points (3, -3) and (-5, 2), we need to determine the values of A, B, and C in the equation Ax + By = C.
First, we need to find the slope of the line. The formula for the slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
Using the coordinates of the given points, we can calculate the slope as follows:
m = (2 - (-3)) / (-5 - 3) = 5 / (-8) = -5/8
Next, we can rewrite the equation using the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. We substitute one of the points into the equation to find the value of b.
Using the point (3, -3):
-3 = (-5/8)(3) + b
-24 = -15 + 8b
8b = -9
b = -9/8
So, the equation of the line in slope-intercept form is:
y = (-5/8)x - (9/8)
To convert the equation to the standard form Ax + By = C, where A, B, and C are integers, we can multiply both sides of the equation by 8 to eliminate fractions:
8y = -5x - 9
But since A must be positive and the greatest common divisor of A, B, and C must be 1, we need to make sure A is positive. We can multiply both sides of the equation by -1 to achieve this:
-8y = 5x + 9
Finally, rearranging the equation to match the standard form, we get:
5x + 8y = -9
Therefore, the values of A, B, and C are 5, 8, and -9, respectively. The sum A + B + C is:
5 + 8 + (-9) = 4
Hence, A + B + C = 4.