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 $\boxed{\textbf{(D) }m\neq 1}$ .

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1975 AHSME Problems/Problem 2

Solution