Final answer:
The standard form of the equation for the line that passes through the points (6, -12) and (10, 11) is -23x + 4y = -252. We find the slope as 23/4 and then use it along with one of the points to solve for the y-intercept. After rearranging the equation into standard form, we multiply by 4 to remove the fraction.
Step-by-step explanation:
To write the standard form of the equation for the line that passes through the points (6, -12) and (10, 11), we first need to find the slope of the line. The slope, m, is found by calculating the rise over run, which is the difference in the y-coordinates divided by the difference in the x-coordinates.
The slope of the line is calculated as follows:
m = (11 - (-12))/(10 - 6) = 23/4.
Next, we can use the slope and one of the given points to write the slope-intercept form of the equation, which is y = mx + b. After substituting the slope and one point, (6, -12), into the equation, we get:
-12 = (23/4)(6) + b, which simplifies to b = -63.
Therefore, the slope-intercept form of the line is:
y = (23/4)x - 63.
To convert this to standard form, Ax + By = C, we multiply both sides of the equation by 4 to eliminate the fraction, and we get:
4y = 23x - 252.
Finally, we rearrange the equation to the standard form:
-23x + 4y = -252. This equation is in standard form and represents the line through the points (6, -12) and (10, 11).