Question
\( \left. \begin{array} { l } { \frac { x - y } { 2 } = \frac { y } { 3 } } \\ { x + y = 16 } \\ { \Rightarrow x \cdot y = ? } \end{array} \right. \)
Ask by Hilton Mckenzie. in Turkey
Feb 19,2025
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Tutor-Verified Answer
Answer
x va y ni hisoblash uchun tenglamalar sistemasini yechib, x = 10 va y = 6 topiladi. Shunda x · y = 60.
Solution
Biz quyidagi tenglamalar sistemasini ko'rib chiqamiz:
1) (x – y)/2 = y/3
2) x + y = 16
Birinchi tenglamani yechish uchun ikki qismning maxrajlarini yo‘q qilish uchun ekvivalent koeffitsient topamiz:
3(x – y) = 2y
ya'ni, 3x – 3y = 2y
Shundan 3x = 5y bo‘ladi, demak: x = (5/3)y
Endi bu x ifodasini ikkinchi tenglamada o‘rniga qo‘yamiz:
(5/3)y + y = 16
(5/3 + 1)y = 16
(5/3 + 3/3)y = (8/3)y = 16
Shu yerda y ni aniqlash uchun har ikki tomonni 3/8 ga ko‘paytiramiz:
y = 16 * (3/8) = 6
Keyinchalik, x ni hisoblaymiz:
x = (5/3) * 6 = 10
Endi x va y ko‘paytmasi:
x ⋅ y = 10 ⋅ 6 = 60
Demak, javob: 60.
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Simplify this solution Mind Expander
To solve the system of equations, we first manipulate the first equation: \[ \frac{x - y}{2} = \frac{y}{3} \] Multiplying both sides by 6 to eliminate the fractions gives: \[ 3(x - y) = 2y \implies 3x - 3y = 2y \implies 3x = 5y \implies y = \frac{3}{5}x \] Now, we substitute \(y\) into the second equation \(x + y = 16\): \[ x + \frac{3}{5}x = 16 \implies \frac{8}{5}x = 16 \implies x = 16 \cdot \frac{5}{8} = 10 \] Now we can find \(y\): \[ y = \frac{3}{5} \cdot 10 = 6 \] Finally, calculate \(x \cdot y\): \[ x \cdot y = 10 \cdot 6 = 60 \] So, the answer is \(60\).
