OG详解-OG6 数学2 Q27

The correct answer is -28. A system of two linear equations in two variables, x and y, has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in the form Ax+By = C, where A, B, and C are constants, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients for x and y in other equation. The first equation in the given system, 48x-64y = 48y+24, can be written in the form Ax+By = C by subtracting 48y from both sides of the equation to yield 48x-112y = 24. The second equation in the given system, ry = 1​8​- 12x, can be written in the form Ax+By = C by adding 12x to both sides of the equation to yield 12x+ry = 1​8​. The coefficient of x in the second equation is times the coefficient of x in the first equation. That is, 48(14​​) = 12. For the lines to be parallel, the coefficient of y in the second equation must also be times the coefficient of y in the first equation. Therefore, -112(14​​) = r, or -28 = r. Thus, if the given system has no solution, the value of r is -28.