Final answer:
To determine the value of k, we use the slope formula given two points on the line and the point-slope form of the line equation using the known y-intercept. After solving for k, we can determine the slope and construct the equation of the line in the form y = mx + b.
Step-by-step explanation:
To find the value of k for the line that passes through the points (3k, 6k - 5) and (-1, -7) and has a y-intercept of -5, we will use the slope formula and the point-slope form of the line equation.
Firstly, the slope of the line can be calculated since we have two points on the line:
m = (y2 - y1) / (x2 - x1)
m = (-7 - (6k - 5)) / (-1 - 3k)
Now we assume the slope m is also the slope when the line crosses the y-axis at the y-intercept, which is -5.
Since we know a point on the y-axis (0, -5), we can use the point-slope form:
y - y1 = m(x - x1)
Using the y-intercept point, the equation becomes:
y + 5 = m(x)
We now have two forms of the equation for slope. The first one is derived from the given points and the second one from the y-intercept. Setting these two expressions for m equal, we can solve for k.
After calculating the value of k, we use it to find the actual slope and then substitute the slope (m) and y-intercept (-5) into the slope-intercept form of the line equation, y = mx + b, to get the final equation of the line.