Find the values of $α + β$ for which the following system of linear equations has infinite number of solutions $2 x + 3 y = 7$ & $2 α x + (α + β) y = 28$

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Solution

For infinite solution $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\Rightarrow \frac{2}{2 α} = \frac{3}{α + β} = \frac{7}{28} \Rightarrow \frac{1}{α} = \frac{3}{α + β} = \frac{1}{4}$ $\Rightarrow α = 4$
[From the first & third term]
$\Rightarrow \frac{3}{4 + β} = \frac{1}{4} \Rightarrow 4 + β = 12 \Rightarrow β = 8$

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α

β

2 x + 3 y = 7

2 α x + (α + β) y = 28

α

β

2 x + 3 y = 7; 2 α x + (α + β) y = 28

α & β

2 α β + (α + β) y = 28

(α, β)

(1 + α) x + β y + z = 2 α x + (1 + β) y + z = 3 α x + β y + 2 z = 2

2 x + 3 y = 7 2 a x + (a + b) y = 28

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