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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