The lines x = 14 y + a, y = 14 x + b intersect at the point (1, 2). What is a + b?

2006 AMC 12B Problem 8

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The lines

x = \tfrac14 y + a, \qquad y = \tfrac14 x + b

intersect at the point

(1, 2).

What is

a + b?

0

\dfrac{3}{4}

1

2

\dfrac{9}{4}

Substitute

x = 1

and

y = 2

into both equations

Solve each equation for

a

and

b

separately, then add

Substituting

(1, 2)

gives

1 = \frac{2}{4} + a \quad\Rightarrow\quad a = \frac{1}{2},

and

2 = \frac{1}{4} + b \quad\Rightarrow\quad b = \frac{7}{4}.

Therefore

a + b = \frac{1}{2} + \frac{7}{4} = \frac{9}{4}.

Thus, the correct answer is E.