1975 AHSME Problems/Problem 2

For which real values of m are the simultaneous equations \begin{align*}y &= mx + 3 \\  y& = (2m - 1)x + 4\end{align*} satisfied by at least one pair of real numbers $(x,y)$?

$\textbf{(A)}\ \text{all }m\qquad \textbf{(B)}\ \text{all }m\neq 0\qquad \textbf{(C)}\ \text{all }m\neq 1/2\qquad \textbf{(D)}\ \text{all }m\neq 1\qquad \\ \textbf{(E)}\ \text{no values of }m$


Solution

Solution by e_power_pi_times_i


Solving the systems of equations, we find that $mx+3 = (2m-1)x+4$, which simplifies to $(m-1)x+1 = 0$. Therefore $x = \dfrac{1}{1-m}$. $x$ is only a real number if $\boxed{\textbf{(D) }m\neq 1}$.

Invalid username
Login to AoPS