Final answer:
To find the value of k for the given line to have a y-intercept of -7, we calculate the slope with the points (-4, k) and (-1, 2k) and use the point-slope form of a line equation. By substituting the y-intercept (-7) and solving for k, we find that the correct value of k is -3.
Step-by-step explanation:
To find the value of k that makes the line passing through (-4, k) and (-1, 2k) have a y-intercept of -7, we need to determine the equation of that line. First, we determine the slope (m) of the line using the two given points.
The formula for slope is m = (y2 - y1) / (x2 - x1), which gives us m = (2k - k) / (-1 - (-4)) = k / 3.
Next, we'll use the point-slope form of the equation of a line, y - y1 = m(x - x1), substituting one of the points and the slope. Using the point (-4, k), we get y - k = (k/3)(x + 4).
Since we want the y-intercept to be -7, we set x to 0 and y to -7 in the equation, resulting in -7 - k = (k/3)(0 + 4). Simplifying this equation by distributing the k/3 to 4 and moving all terms to one side gives us -7 - k = (4k/3). Multiplying every term by 3 to get rid of the fraction gives us -21 - 3k = 4k. Adding 3k to both sides results in -21 = 7k, yielding k = -3.
Therefore, the correct value of k is -3 (Option A).