How do you find A and B if the slope of coefficients of x and y in Ax+By-13=0 is 6 and the slope of the line is 7?

Question

The sum of coefficients of x and y in Ax+By+C=0 is 6. If the slope of the line is 7, find A and B. How do I find it? What should I do first?

Answer 1

Consider Ax + By = C. Since C does not contribute to the slope of the line, we might as well consider

Ax + By = 0, whence By = -Ax, showing y = - (A/B)x, showing the slope is -A/B for any line Ax + By = C.

In this case, you have Ax + By = 13, where A + B = 6, and -A/B = 7, showing A/B = -7, showing A = -7B. Substitute -7B for A in the equation A + B = 6 to get -7B + B = 6 which you can solve to find B, and then back substitute into A + B = 6 to find A.