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

1975 AMC 12 Problem 2

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For which real values of

m

are the simultaneous equations

\begin{aligned} y&=mx+3,\\ y&=(2m-1)x+4 \end{aligned}

satisfied by at least one pair of real numbers

(x,y)?

all

m

all

m\ne0

all

m\ne\frac12

all

m\ne1

no values of

m

Two nonvertical lines fail to meet only when they are parallel and distinct

Set the slopes

m

and

2m-1

equal

The lines fail to intersect only if

m=2m-1,

which gives

m=1.

Their

y

-intercepts are

3

and

4,

so at that value they are distinct parallel lines. Every other

m

gives one intersection.

Therefore, the correct answer is D.