48 x - 64 y = 48 y + 24 r y = 1 8 - 12 x In the

Question

48 x - 64 y = 48 y + 24 r y = 1 8 - 12 x In the given system of equations, r is a constant. If the system has no solution, what is the value of r ?

Accepted Answer

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 A x + B y = 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 the other equation. The first equation in the given system, 48 x - 64 y = 48 y + 24 , can be written in the form A x + B y = C by subtracting 48 y from both sides of the equation to yield 48 x - 112 y = 24 . The second equation in the given system, r y = 1 8 - 12 x , can be written in the form A x + B y = C by adding 12 x to both sides of the equation to yield 12 x + r y = 1 8 . The coefficient of x in the second equation is 1 4 times the coefficient of x in the first equation. That is, 48 1 4 = 12 . For the lines to be parallel, the coefficient of y in the second equation must also be 1 4 times the coefficient of y in the first equation. Therefore, - 112 1 4 = r , or - 28 = r . Thus, if the given system has no solution, the value of r is - 28 .

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