OG详解-OG3 数学2 Q27

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, Ax + By = C and Dx + Ey = F , are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation, meaning DA​=B​E​; and the lines are distinct if the constants are not proportional, meaningF​C​is not equal to DA​​ or EB​​. 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 18y -3x = 8-18y . Adding 18y to each side of this equation yields 36y -3x = 8, or -3x + 36y = 8. The second equation in the given system is 1​2​x + 3​2​= py + 9​2​. Multiplying each side of this equation by 2 yields x + 3 = 2py + 9. Subtracting 2py from each side of this equation yields x + 3-2py = 9. Subtracting 3 from each side of this equation yields x -2py = 6. Therefore, the two equations in the given system, written in standard form, are -3x + 36y = 8 and x -2py = 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​ = −2p​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.