Final answer:
The equation of the line through the given points is y = (3/13)x + 47/13. The value of k is 59/13.
Step-by-step explanation:
To find the equation of a line perpendicular to another line, we need to determine the slope of the given line and then take the negative reciprocal of that slope. Let's determine the slope of the line through the points (-3,6) and (0,-7). The slope is given by the formula: slope = (y2 - y1) / (x2 - x1). Substituting the coordinates, we have: slope = (-7 - 6) / (0 - (-3)) = -13/3.
Since the line we are looking for is perpendicular to this line, the slope of the new line will be the negative reciprocal of -13/3. The negative reciprocal is 3/13.
Now, we can use the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept. We can substitute one of the given points, (3,4), into this equation to find the value of b. Substituting the coordinates, we have: 4 = (3/13)(3) + b. Solving for b, we get: b = 4 - 9/13 = 47/13.
Therefore, the equation of the line through the points (3,4) and (4,k) that is perpendicular to the line through the points (-3,6) and (0,-7) is y = (3/13)x + 47/13.
To find the value of k, we substitute the x-coordinate of the second point, 4, into the equation. Substituting, we have: k = (3/13)(4) + 47/13 = 12/13 + 47/13 = 59/13.
Therefore, k = 59/13.