Solve the following pair of equations by cross- multiplication method:
$2 a x + 3 b y = a + 2 b$
$3 a x + 2 b y = 2 a + b$

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Solution

$2 a x + 3 b y = a + 2 b$
$3 a x + 2 b y = 2 a + b$
Give, pair of equations is
$2 a x + 3 b y = a + 2 b \Rightarrow 2 a x + 3 b y - (a + 2 b) = 0$
$3 a x + 2 b y = 2 a + b \Rightarrow 3 a x + 2 b y - (2 a + b) = 0$
By cross-multiplication method, we have
$\Rightarrow \frac{x}{- 3 a b (2 a + b) + 2 b (a + 2 b)} = \frac{y}{- 3 a (a + 2 b) + 2 a (2 a + b)} = \frac{1}{4 a b - 9 a b}$
$\Rightarrow \frac{x}{- 4 b a - 3 b^{2} + 4 a b + b^{2}} = \frac{y}{- 3 a^{2} - 6 b a + 4 a^{2} + 2 a b} = \frac{1}{- 5 a b}$
$\Rightarrow \frac{x}{b^{2} - 4 a b} = \frac{y}{- 4 a b + a^{2}} = \frac{1}{- 5 a b}$
I II III
On taking I and III ratio, we get
$\Rightarrow \frac{x}{b^{2} - 4 a b} = \frac{1}{- 5 a b}$
$\Rightarrow x = \frac{b (b - 4 a)}{5 a}$
$\Rightarrow \frac{4 a - b}{5 a}$
On taking II and III ratio, we get
$\Rightarrow \frac{y}{- 4 a b + a^{2}} = \frac{1}{- 5 a b}$
$\Rightarrow y = \frac{a (a - 4 b)}{- 5 a b}$
$\Rightarrow y = \frac{4 b - a}{5 b}$

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Similar questions

Solve each of the following systems of equations by using the method of cross multiplication:

$2 a x + 3 b y = (a + 2 b), 3 a x + 2 b y = (2 a + b)$

2 a x + 3 b y = (a + 2 b), 3 a x + 2 b y = (2 a + b) .

2 (a x - b y) + a + 4 b = 0 2 (b x + a y) + b - 4 a = 0

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