Final answer:
To check if additional points are on a line, calculate the equation of the line first then substitute the points' coordinates into the equation. If the equation holds, the point is on the line. Using this method, we can confirm that the point (1, 5) is not on the line that includes (-5,14) and (5,-6).
Step-by-step explanation:
To determine whether each point is on the line that contains (-5, 14) and (5, -6), we must first find the equation of the line. This can be done by calculating the slope and using one of the points to find the y-intercept.
The slope m is calculated by the change in y over the change in x (rise over run):
m = (y2 - y1) / (x2 - x1)
= (-6 - 14) / (5 - (-5))
= -20 / 10
= -2
Thus, we have a straight line with negative slope. Using the point-slope form of a line equation, y - y1 = m(x - x1), and one of the points (e.g., (5, -6)), we get:
y + 6 = -2(x - 5)
This simplifies to:
y = -2x + 4
Now, we can test each of the provided points by substituting the x and y values into our equation to see if the equation holds true.
For the point (1, 5):
5 = -2(1) + 4
5 ≠ -2 + 4
5 ≠ 2
This point is not on the line because the equation does not hold true.
By performing a similar substitution for the other points, we can determine if they lie on the line as well. If the left-hand side equals the right-hand side after substitution, the point is on the line; otherwise, it is not.