7 8 y - 5 8 x = 4 7 - 7 8 y 5 4 x + 7 4

Question

7 8 y - 5 8 x = 4 7 - 7 8 y 5 4 x + 7 4 = p y + 15 4 In the given system of equations, p is a constant. If the system has no solution, what is the value of p ?

Accepted Answer

The correct answer is 7 2 . 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. Two lines represented by equations in standard 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 in the other equation. The first equation in the given system, 7 8 y - 5 8 x = 4 7 - 7 8 y , can be written in standard form by adding 7 8 y to both sides of the equation, which yields 14 8 y - 5 8 x = 4 7 , or - 5 8 x + 14 8 y = 4 7 . Multiplying each term in this equation by - 8 yields 5 x - 14 y = - 32 7 . The second equation in the given system, 5 4 x + 7 4 = p y + 15 4 , can be written in standard form by subtracting 7 4 and p y from both sides of the equation, which yields 5 4 x - p y = 8 4 . Multiplying each term in this equation by 4 yields 5 x - 4 p y = 8 . The coefficient of x in the first equation, 5 x - 14 y = - 32 7 , is equal to the coefficient of x in the second equation, 5 x - 4 p y = 8 . For the lines to be parallel, and for the coefficients for x and y in one equation to be proportional to the corresponding coefficients in the other equation, the coefficient of y in the second equation must also be equal to the coefficient of y in the first equation. Therefore, - 14 = - 4 p . Dividing both sides of this equation by - 4 yields - 14 - 4 = p , or p = 7 2 . Therefore, if the given system of equations has no solution, the value of p is 7 2 . Note that 7/2 and 3.5 are examples of ways to enter a correct answer.

Question 7-8-y-