+16 Write the equation of the line below in the form A(x) + B(x) = C, where A, B, and C are integers with greatest common divisor 1, and A is positive.
The line consists of two known points: (-4, 2) and (1, -1)
+1133 Finding the Slope (m):
The slope (m) of the line passing through points (-4, 2) and (1, -1) can be found using the following formula:
m = (y2 - y1) / (x2 - x1)
Here, (x1, y1) = (-4, 2) and (x2, y2) = (1, -1)
m = (-1 - 2) / (1 - (-4)) = -3 / 5
Finding the y-intercept (b):
We can use the point-slope form of the equation to find the y-intercept (b). Here, we can use either of the given points. Let's use (-4, 2).
y = mx + b
2 = (-3/5) * (-4) + b
2 = 12/5 + b
b = 2 - 12/5 = -2/5
Writing the Equation in Ax + By = C form:
Now, we can express the equation in the desired form:
A(x) + B(y) = C
Substitute the slope (m) and y-intercept (b):
A(x) + (-3/5)(y) = -2/5
Finding A, B, and C with GCD = 1 and A positive:
To ensure the greatest common divisor (GCD) of A, B, and C is 1 and A is positive, we can multiply the entire equation by the least common multiple (LCM) of the denominators of m and b (which is 5 in this case).
5A(x) - 3y = -2
Since A is positive, we can simply set A = 5:
5(x) - 3y = -2
Therefore, the equation of the line in the desired form is:
5x - 3y = -2
