4 x - 6 y = 10 y + 2 t y = 1 2 + 2 x In the

Question

4 x - 6 y = 10 y + 2 t y = 1 2 + 2 x In the given system of equations, t is a constant. If the system has no solution, what is the value of t ?

Accepted Answer

The correct answer is 8 . The given system of equations can be solved using the elimination method. Multiplying both sides of the second equation in the given system by - 2 yields - 2 t y = - 1 - 4 x , or - 1 - 4 x = - 2 t y . Adding this equation to the first equation in the given system, 4 x - 6 y = 10 y + 2 , yields 4 x - 6 y + - 1 - 4 x = 10 y + 2 + - 2 t y , or - 1 - 6 y = 10 y - 2 t y + 2 . Subtracting 10 y from both sides of this equation yields - 1 - 6 y - 10 y = 10 y - 2 t y + 2 - 10 y , or - 1 - 16 y = - 2 t y + 2 . If the given system has no solution, then the equation - 1 - 16 y = - 2 t y + 2 has no solution. If this equation has no solution, the coefficients of y on each side of the equation, - 16 and - 2 t , must be equal, which yields the equation - 16 = - 2 t . Dividing both sides of this equation by - 2 yields 8 = t . Thus, if the system has no solution, the value of t is 8 . Alternate approach: 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 the form A x + B y = C , where A , B , and C are constant terms, are parallel if the ratio of the x -coefficients is equal to the ratio of the y -coefficients, and distinct if the ratio of the x -coefficients are not equal to the ratio of the constant terms. Subtracting 10 y from both sides of the first equation in the given system yields 4 x - 6 y - 10 y = 10 y + 2 - 10 y , or 4 x - 16 y = 2 . Subtracting 2 x from both sides of the second equation in the given system yields t y - 2 x = 1 2 + 2 x - 2 x , or - 2 x + t y = 1 2 . The ratio of the x -coefficients for these equations is - 2 4 , or - 1 2 . The ratio of the y -coefficients for these equations is - t 16 . The ratio of the constant terms for these equations is 1 2 2 , or 1 4 . Since the ratio of the x -coefficients, - 1 2 , is not equal to the ratio of the constants, 1 4 , the lines represented by the equations are distinct. Setting the ratio of the x -coefficients equal to the ratio of the y -coefficients yields - 1 2 = - t 16 . Multiplying both sides of this equation by - 16 yields - 1 2 - 16 = - t 16 - 16 , or t = 8 . Therefore, when t = 8 , the lines represented by these equations are parallel. Thus, if the system has no solution, the value of t is 8 .

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