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Question

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


Solution

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

1796091_1816909_ans_8035e7983f0a44588a59cf6841ff0b1f.PNG

Mathematics

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