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A line passes through the points (-3,-5) and (6,1). The equation of this line can be written in the form Ax + By = C, where A, B, and C are integers with greatest common divisor 1, and A is positive. Find A + B+ C.

 Mar 18, 2020
 #1
avatar+485 
+2

Let's look at the two points we're given: 

 

we can find the slope of the line by taking the difference of the respective ys, and dividing by the difference between the respective x's(rise over run). This gives us a slope of : 1 - (-5) / 6 - (-3), or (1+5) / (6+3), which is equal to 6/9 or 2/3. 

 

With this, we can write the equation of the line as:

 

y = 2/3x + b. Now, we just have to solve for b, which we can find by just plugging in a point on that line. Let's use the point (6,1).

 

Plugging in (6,1) for the x and y values respectively, we get:

 

1 = 6 * 2/3 + b, or

1 = 4 + b. Subtracting 4 on both sides, we get:

-3 = b

 

This then gives us the equation y = 2/3x - 3, but since the question asks for standard form which is Ax + by + c = 0, we can rewrite the equation as:

 

3y = 2x - 9 from multiplying the equation by 3.

 

Then, rearranging to get our desired form, we get :

 

2x - 3y - 9 = 0. Since the question asks for A + B + C, we get 2 + (-3) + (-9), which is equal to 2-12, or -10

 Mar 18, 2020

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