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System of Equations with Absolute Values

Source: 2021 Fall AMC 10A 14

November 11, 2021
AMCAMC 10AMC 10 Aalgebrasystem of equationsabsolute value

Problem Statement

How many ordered pairs (x,y)(x,y) of real numbers satisfy the following system of equations? \begin{align*} x^2+3y&=9\\ (|x|+|y|-4)^2&=1\\ \end{align*} <spanclass=latexbold>(A)</span>1<spanclass=latexbold>(B)</span>2<spanclass=latexbold>(C)</span>3<spanclass=latexbold>(D)</span>5<spanclass=latexbold>(E)</span>7<span class='latex-bold'>(A)</span>\: 1\qquad<span class='latex-bold'>(B)</span> \: 2\qquad<span class='latex-bold'>(C)</span> \: 3\qquad<span class='latex-bold'>(D)</span> \: 5\qquad<span class='latex-bold'>(E)</span> \: 7