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# Cartesian plane question

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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)

Apr 8, 2024
edited by Seraphspace  Apr 8, 2024

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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

Apr 14, 2024