दो चर वाले रैखिक समीकरण युग्म

Question

निम्न समीकरणों के युग्मों के रैखिक समीकरणों के युग्म में बदल करके हल कीजिए:

$fraction numerator 1 over denominator 3 x italic space plus space y end fraction space plus space fraction numerator 1 over denominator 3 x italic space minus space y end fraction space equals space 3 over 4 fraction numerator 1 over denominator 2 space left parenthesis space 3 x italic space plus space y italic space right parenthesis end fraction space plus space fraction numerator 1 over denominator 2 space left parenthesis space 3 x italic space minus space y italic space right parenthesis end fraction space equals space fraction numerator negative space 1 over denominator space space 8 end fraction$

Solution

म ा न ा space space fraction numerator 1 over denominator 3 straight x space plus space straight y end fraction space equals space straight u space और space space fraction numerator 1 over denominator 3 straight x space minus space straight y end fraction space equals space straight v

straight u space plus space straight v space equals space 3 over 4 space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis straight i right parenthesis straight u space minus space straight v space equals space fraction numerator negative space 1 over denominator 4 end fraction space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis ii right parenthesis

समीकरण (i) और (ii) को जोड़ने पर

2 straight u space equals space 3 over 4 space plus space open parentheses space fraction numerator negative space 1 over denominator 4 end fraction space close parentheses 2 straight u space equals space fraction numerator 3 space minus space 1 over denominator 4 end fraction space equals space 2 over 4 rightwards double arrow space space space space space straight u space equals space 1 fourth

u का मान समीकरण (i) में रखने पर

1 fourth space plus space straight v space equals space 3 over 4 straight v space equals space 3 over 4 space minus space 1 fourth space equals space fraction numerator 3 space minus space 1 over denominator 4 end fraction space equals space 2 over 4 space equals space 1 half fraction numerator 1 over denominator 3 straight x space plus space straight y end fraction space equals space 1 fourth comma space space space fraction numerator 1 over denominator 3 straight x space minus space straight y end fraction space equals space 1 half 3 straight x space plus space straight y space equals space 4 space space space space space... left parenthesis iii right parenthesis 3 straight x space minus space straight y space equals space 2 space space space space space... left parenthesis iv right parenthesis

समीकरण (iii) और (iv) को जोड़ने पर

6x = 6
⇒ x = 1
x का मान समीकरण (iii) में रखने पर
3 x + y = 4
⇒ 3(1) + y = 4
⇒ y = 1
x = 1, y = 1