Let c be a constant. The simultaneous equations x-y&=2, cx+y&=3. have a solution (x,y) inside Quadrant I if and only if

1987 AMC 12 Problem 11

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11.

Let

c

be a constant. The simultaneous equations

\begin{aligned} x-y&=2,\\ cx+y&=3. \end{aligned}

have a solution

(x,y)

inside Quadrant I if and only if

c=-1

c\gt-1

c\lt\frac32

0\lt c\lt\frac32

-1\lt c\lt\frac32

Solve the two equations for

x

and

y

in terms of

c

Quadrant I requires both coordinates to be strictly positive

Solving the system gives

x=\frac5{c+1},\qquad y=\frac{3-2c}{c+1}.

The condition

x\gt0

requires

c\gt-1.

With this positive denominator,

y\gt0

is equivalent to

c\lt\frac{3}{2}.

Hence

-1\lt c\lt\frac{3}{2}.

Thus the correct answer is E.