3 2 y - 1 4 x = 2 3 - 3 2 y 1 2 x + 3 2

Question

3 2 y - 1 4 x = 2 3 - 3 2 y 1 2 x + 3 2 = p y + 9 2 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 6 . 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 parallel and distinct. Lines represented by equations in standard form, A x + B y = C and D x + E y = F , are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation, meaning D A = E B ; and the lines are distinct if the constants are not proportional, meaning F C is not equal to D A or E B . The first equation in the given system is 3 2 y - 1 4 x = 2 3 - 3 2 y . Multiplying each side of this equation by 12 yields 18 y - 3 x = 8 - 18 y . Adding 18 y to each side of this equation yields 36 y - 3 x = 8 , or - 3 x + 36 y = 8 . The second equation in the given system is 1 2 x + 3 2 = p y + 9 2 . Multiplying each side of this equation by 2 yields x + 3 = 2 p y + 9 . Subtracting 2 p y from each side of this equation yields x + 3 - 2 p y = 9 . Subtracting 3 from each side of this equation yields x - 2 p y = 6 . Therefore, the two equations in the given system, written in standard form, are - 3 x + 36 y = 8 and x - 2 p y = 6 . As previously stated, if this system has no solution, the lines represented by the equations in the xy -plane are parallel and distinct, meaning the proportion 1 - 3 = - 2 p 36 , or - 1 3 = - p 18 , is true and the proportion 6 8 = 1 - 3 is not true. The proportion 6 8 = 1 - 3 is not true. Multiplying each side of the true proportion, - 1 3 = - p 18 , by - 18 yields 6 = p . Therefore, if the system has no solution, then the value of p is 6 .

Question 3-2-y-