Question : The graphs of the linear equations $3x-2y=8$ and $4x+3y=5$ intersect at the point ${P}( \alpha, \beta)$. What is the value of $(2 \alpha-\beta)$?
Option 1: 3
Option 2: 4
Option 3: 6
Option 4: 5
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Correct Answer: 5
Solution : $3x-2y=8$ ------------(1) $4x+3y=5$ ---------------------(2) Solve this system of equations. Rearranging the first equation, $⇒y = \frac{3}{2}x - 4$ Substituting this into the second equation, $⇒4x + 3(\frac{3}{2}x - 4) = 5$ $⇒4x +\frac{9}{2}x - 12 = 5$ $⇒17x-24=10$ $⇒17x=34$ $⇒x = 2$ Substituting $x = 2$ into the first equation, $⇒y = \frac{3}{2}\times 2 - 4 = 3-4 =-1$ So, the intersection point is $P(2, -1)$, which means $\alpha = 2$ and $\beta = -1$. Therefore, $⇒2 \alpha - \beta = 2 \times 2 - (-1) = 4 + 1 =5$ Hence, the correct answer is 5.
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