Final answer:
Finding the value of k involves using the slope formula and the given points, which leads to the determination that k=2 and the equation of the line is y = 2x - 5.
Step-by-step explanation:
Given that a line passes through the points (3k,6k-5) and (-1,-7) and has a y-intercept of -5, we can use these conditions to find the value of k and the equation of the line. As the y-intercept is -5, this will be the value of 'b' in the line's equation of the form y = mx + b.
Firstly, to find the slope (m), we use the formula for the slope of a line through two points (x1,y1) and (x2,y2): m = (y2 - y1) / (x2 - x1). Plugging the given points in, we get: m = (-7 - (6k - 5)) / (-1 - 3k). The slope m is also the coefficient of x in the equation y = mx - 5, since the line passes through the y-axis at -5, which is the y-intercept.
Using the given points, we can find that m = (2 - 6k) / (4k + 1). Since the line must have the same slope everywhere, we set this equal to the slope from y = mx - 5. Solving for k, we find k=2, which gives us the slope of 2. Thus, the equation of the line is y = 2x - 5.