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If 4x = p(x + 3) + q(x - 1) is an identity, then the values of p and q are?

11, -3
21, 3
31, 1
43, 1

Answer:- 1

Explanation:-

Solution:

\begin{aligned} 4 x = p (x + 3) + q (x - 1) \\ \Rightarrow 4 x = p x + 3 p + q x - q \\ \Rightarrow 4 x = (p + q) x + (3 p - q) \\ \Rightarrow p + q = 4 a n d 3 p - q = 0 \\ \Rightarrow p + q = 4 a n d p = \frac{q}{3} \\ \Rightarrow \frac{q}{3} + q = 4 \\ \Rightarrow \frac{4 q}{3} = 4 \\ \Rightarrow q = 3 \\ p + q = 4 a n d \\ \Rightarrow p = 1 \\ \Rightarrow q = 3 \end{aligned}

$\eqalign{ & {\text{4}}x = p\left( {x + 3} \right) + q\left( {x - 1} \right) \cr & \Rightarrow 4x = px + 3p + qx - q \cr & \Rightarrow 4x = \left( {p + q} \right)x + \left( {3p - q} \right) \cr & \Rightarrow p + q = 4\,and\,3p - q = 0 \cr & \Rightarrow p + q = 4\,and\,p = \frac{q}{3} \cr & \Rightarrow \frac{q}{3} + q = 4 \cr & \Rightarrow \frac{{4q}}{3} = 4 \cr & \Rightarrow q = 3 \cr & p + q = 4\,\,\,and \cr & \Rightarrow p = 1 \cr & \Rightarrow q = 3 \cr}$

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", "dateCreated": "7/24/2019 10:09:12 AM", "author": { "@type": "Person", "name": "Nitin Sir" } }, "suggestedAnswer": { "@type": "Answer", "text": "

Solution:

\begin{aligned} 4 x = p (x + 3) + q (x - 1) \\ \Rightarrow 4 x = p x + 3 p + q x - q \\ \Rightarrow 4 x = (p + q) x + (3 p - q) \\ \Rightarrow p + q = 4 a n d 3 p - q = 0 \\ \Rightarrow p + q = 4 a n d p = \frac{q}{3} \\ \Rightarrow \frac{q}{3} + q = 4 \\ \Rightarrow \frac{4 q}{3} = 4 \\ \Rightarrow q = 3 \\ p + q = 4 a n d \\ \Rightarrow p = 1 \\ \Rightarrow q = 3 \end{aligned}

$\eqalign{ & {\text{4}}x = p\left( {x + 3} \right) + q\left( {x - 1} \right) \cr & \Rightarrow 4x = px + 3p + qx - q \cr & \Rightarrow 4x = \left( {p + q} \right)x + \left( {3p - q} \right) \cr & \Rightarrow p + q = 4\,and\,3p - q = 0 \cr & \Rightarrow p + q = 4\,and\,p = \frac{q}{3} \cr & \Rightarrow \frac{q}{3} + q = 4 \cr & \Rightarrow \frac{{4q}}{3} = 4 \cr & \Rightarrow q = 3 \cr & p + q = 4\,\,\,and \cr & \Rightarrow p = 1 \cr & \Rightarrow q = 3 \cr}$ ", "dateCreated": "7/24/2019 10:09:12 AM" } }

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