If-x-y-satisfies-the-system-of-equations-x-x-y-2-0-y-y-5x-1-find-the-value-of-x-y-

Question Number 116601 by ZiYangLee last updated on 05/Oct/20

If (x, y) satisfies the system of equations

{\begin{cases} ∣ x ∣ - x - y + 2 = 0 \\ ∣ y ∣ + y + 5 x = 1 \end{cases}

find the value of x + y .

Answered by mr W last updated on 05/Oct/20

{\begin{cases} x - x - y + 2 = 0 \\ y + y + 5 x = 1 \end{cases}

\Rightarrow y = 2, x = - \frac{3}{5} n o t ⩾ 0 \Rightarrow n o s o l u t i o n

{\begin{cases} x - x - y + 2 = 0 \\ - y + y + 5 x = 1 \end{cases}

\Rightarrow y = 2 n o t < 0 \Rightarrow n o s o l u t i o n

{\begin{cases} - x - x - y + 2 = 0 \Rightarrow 2 x + y = 2 \\ y + y + 5 x = 1 \Rightarrow 5 x + 2 y = 1 \end{cases}

\Rightarrow x = - 3, y = 8 \Rightarrow s o l u t i o n

{\begin{cases} - x - x - y + 2 = 0 \Rightarrow 2 x + y = 2 \\ - y + y + 5 x = 1 \end{cases}

\Rightarrow x = \frac{1}{5} n o t < 0 \Rightarrow n o s o l u t i o n

s o l u t i o n :

x = - 3, y = 8

\Rightarrow x + y = 5

Commented by ZiYangLee last updated on 05/Oct/20

Excellent ★ ★

Answered by bemath last updated on 05/Oct/20

{\begin{cases} ∣ x ∣= x + y - 2 \\ ∣ y ∣= 1 - 5 x - y \end{cases}

\to∣ x ∣ + ∣ y ∣= - 4 x - 1 \dots (iii)

case (1) \to {\begin{cases} x ⩾ 0 \\ y ⩾ 0 \end{cases} \Rightarrow x + y = - 4 x - 1

y = - 5 x - 1 \land 2 y = 1 - 5 x

substitute \Rightarrow 2 (- 5 x - 1) = 1 - 5 x

\Rightarrow - 10 x - 2 = 1 - 5 x

\Rightarrow - 3 = 5 x; x = - \frac{3}{5} \leftarrow rejected

case (2) \to {\begin{cases} x < 0 \\ y ⩾ 0 \end{cases} \Rightarrow - x + y = - 4 x - 1

y = - 3 x - 1 \land 2 y = 1 - 5 x

\Rightarrow - 6 x - 2 = 1 - 5 x; - x = 3, x = - 3 \leftarrow acceptable

then y = 9 - 1 = 8

therefore x + y = 5

Commented by bobhans last updated on 05/Oct/20

waw \dots . . funtastic . .

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