1969 AMC 12 Problem 18

Question

The number of points common to the graphs of

(x-y+2)(3x+y-4)=0

and

(x+y-2)(2x-5y+7)=0

is:

2

4

6

16

infinite

Each factored equation represents a pair of lines

Intersect each line from the first pair with each line from the second pair and check distinctness

The first graph is the pair

x-y+2=0,

and

3x+y-4=0,

and the second is

x+y-2=0,

and

2x-5y+7=0.

Each line in the first pair meets each line in the second pair. Solving the four pairings gives four distinct points:

\begin{gathered} (0,2),\quad(-1,1),\\ (1,1),\quad\left(\frac{13}{17},\frac{29}{17}\right). \end{gathered}

Thus there are

4

common points.

Therefore, the correct answer is B.

Problem 18 in other years