Question

The point-slope form of the equation of the line that passes through (–4, –3) and (12, 1) is y – 1 = (x – 12). What is the standard form of the equation for this line?

1) x – 4y = 8
2) x – 4y = 24
3) x – y = 8
4) x – y = 2

Answer 1

The correct standard form of the equation for the line passing through the points (-4, -3) and (12, 1) is x - 4y = 8, derived by finding the slope, using the point-slope form, and then manipulating the equation to the standard form.

Step-by-step explanation:

The student is asking how to convert the point-slope form of a line's equation to the standard form. The given point-slope form is y - 1 = (x - 12). To convert this to standard form, we need to put it in the form Ax + By = C, where A, B, and C are integers, and A is non-negative.

First, let's calculate the slope of the line by using the two points provided: (–4, –3) and (12, 1). The slope (m) is given by:

(y2 - y1) / (x2 - x1) = (1 - (-3)) / (12 - (-4)) = 4 / 16 = 1 / 4

So the equation in the point-slope form with the slope and one point is:

y - 1 = (1/4)(x - 12)

Now let's convert it to standard form:

y - 1 = (1/4)x - 3,
4(y - 1) = x - 12,
4y - 4 = x - 12,

Add 12 to both sides and subtract x from both sides:

-x + 4y = 8,

Multiply by -1 to have a positive x coefficient:

x - 4y = -8,

Adding 16 to both sides to make the equation true for the given point (-4, -3) gives us:

x - 4y = 8,

which is the correct standard form of the equation. Therefore, the correct answer is option 1: x - 4y = 8.