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Write the equation of the line below in the form Ax+By=C,  where A, B, and C are integers with greatest common divisor 1 and A is positive.

FYI: The line passes through the points (-4,2) and (1,-1). The slope, which I just figured out, is 3/-5.

PLEASE HELP! ^_^

 

 Jun 1, 2024
edited by Algebro  Jun 1, 2024
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You already found that the slope of the line is -3/5. We can use one of the points and the slope to complete the equation in Ax + By = C form.

 

Let's use the point (-4, 2).

 

Slope-Intercept Form: We know the slope (m) is -3/5 and the point is (-4, 2) represented as (x1, y1). We can use the slope-intercept form to find the y-intercept (b):

 

y = mx + b y1 = m * x1 + b

 

Plugging in the values:

 

2 = (-3/5) * (-4) + b 2 = 12/5 + b

 

Solving for b:

 

b = 2 - 12/5 b = -2/5

 

Therefore, the equation in slope-intercept form is:

 

y = (-3/5)x - 2/5

 

Converting to Ax + By = C: To get the equation in Ax + By = C form, we need to manipulate the slope-intercept form:

 

y = (-3/5)x - 2/5 5y = -3x - 2 (multiply both sides by 5 to eliminate the fraction)

 

Now, the equation is in the form Ax + By = C, where A = -3, B = 5, and C = -2.

 

Greatest Common Divisor (GCD): The greatest common divisor (GCD) of -3, 5, and -2 is 1. However, A needs to be positive according to the prompt.

 

Making A positive: We can multiply the entire equation by -1 to make A positive:

 

(-1) * (5y) = (-1) * (-3x) + (-1) * (-2)

 

5y = 3x + 2

 

3x - 5y = -2

 

Therefore, the standard form of the line is 3x - 5y = -2.

 Jun 1, 2024

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