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Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, then
- a)ad – bc = ±1
- b)ad – bc = - 1
- c)ad – bc = 1
- d)ad - bc need not be equal to ±1.
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Let a, b, c, d, u, v be integers. If the system of equations, a x + b ...
ax + by = u , cx +dy = v ,
since the solution is unique in integers.
Most Upvoted Answer
Let a, b, c, d, u, v be integers. If the system of equations, a x + b ...
We can rewrite the system of equations as follows:
$$\begin{cases}ax + by = u \\ cx + dy = v\end{cases}$$
If we solve for $x$ in terms of $y$ in the first equation, we get:
$$x = \frac{u - by}{a}$$
Substituting this into the second equation, we get:
$$c \cdot \frac{u - by}{a} + dy = v$$
Simplifying this, we get:
$$u\left(\frac{c}{a}\right) + b\left(\frac{-c}{a}\right)y + dy = v$$
Multiplying both sides by $a$, we get:
$$uc - bc y + ady = av$$
Rearranging and factoring, we get:
$$(ad - bc)y = uc - av$$
Since we are given that the system has a unique solution in integers, it follows that $ad - bc \neq 0$. Therefore, we can solve for $y$:
$$y = \frac{uc - av}{ad - bc}$$
Since $y$ is an integer, it follows that $ad - bc$ divides $uc - av$. Therefore, we have shown that:
$$ad - bc \mid uc - av$$
which is equivalent to:
$$ad - bc \mid av - uc$$
This proves option $(\text{C})$.
$$\begin{cases}ax + by = u \\ cx + dy = v\end{cases}$$
If we solve for $x$ in terms of $y$ in the first equation, we get:
$$x = \frac{u - by}{a}$$
Substituting this into the second equation, we get:
$$c \cdot \frac{u - by}{a} + dy = v$$
Simplifying this, we get:
$$u\left(\frac{c}{a}\right) + b\left(\frac{-c}{a}\right)y + dy = v$$
Multiplying both sides by $a$, we get:
$$uc - bc y + ady = av$$
Rearranging and factoring, we get:
$$(ad - bc)y = uc - av$$
Since we are given that the system has a unique solution in integers, it follows that $ad - bc \neq 0$. Therefore, we can solve for $y$:
$$y = \frac{uc - av}{ad - bc}$$
Since $y$ is an integer, it follows that $ad - bc$ divides $uc - av$. Therefore, we have shown that:
$$ad - bc \mid uc - av$$
which is equivalent to:
$$ad - bc \mid av - uc$$
This proves option $(\text{C})$.
Community Answer
Let a, b, c, d, u, v be integers. If the system of equations, a x + b ...
C
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Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, thena)ad – bc =±1b)ad – bc = - 1c)ad – bc = 1d)ad - bc need not be equal to±1.Correct answer is option 'D'. Can you explain this answer?
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Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, thena)ad – bc =±1b)ad – bc = - 1c)ad – bc = 1d)ad - bc need not be equal to±1.Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, thena)ad – bc =±1b)ad – bc = - 1c)ad – bc = 1d)ad - bc need not be equal to±1.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, thena)ad – bc =±1b)ad – bc = - 1c)ad – bc = 1d)ad - bc need not be equal to±1.Correct answer is option 'D'. Can you explain this answer?.
Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, thena)ad – bc =±1b)ad – bc = - 1c)ad – bc = 1d)ad - bc need not be equal to±1.Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, thena)ad – bc =±1b)ad – bc = - 1c)ad – bc = 1d)ad - bc need not be equal to±1.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, thena)ad – bc =±1b)ad – bc = - 1c)ad – bc = 1d)ad - bc need not be equal to±1.Correct answer is option 'D'. Can you explain this answer?.
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