The lines x = \tfrac14 y + a \quad and \quad y = \tfrac14 x + b intersect at the point (1, 2). What is a + b?

2006 AMC 12B Problem 8 — Solution

2006 AMC 12B Problem 8

Below is the professionally curated solution for Problem 8 of the 2006 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 12B solutions, or check the answer key.

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Concepts:system of equations substitution

Difficulty rating: 1250

The lines $x = \tfrac14 y + a \quad \text{and} \quad y = \tfrac14 x + b$ intersect at the point $(1, 2).$ What is $a + b?$

$0$

$\dfrac{3}{4}$

$1$

$2$

$\dfrac{9}{4}$

Solution:

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.

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