Question

solve the following equations by using the methods of substitution. 2x/a+y/b=2 x/a-y/b=4​

Answer 1

Step-by-step explanation:

2x+ by =2,

ax − by =4

2bx+ay=2ab, bx−ay=4ab

Add the two equations,

3bx=6ab

x=2a

Now, 2b(2a)+ay=2ab

ay=−2ab

y=−2b

Answer 2

Appropriate Question :-

Solve the following equations by using the method substitution : $\rm{ \frac{2x}{a} + \frac{y}{b} = 2 }$ and $\rm{ \frac{x}{a} - \frac{y}{b} = 4}$.

Solution :-

Cross multiplication of equation (1)

$\sf:\longmapsto{ \frac{2x}{a} + \frac{y}{b} = 2 } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{2xb + ay = 2} \: \: \: \: \: \: \: \: ... \{eq {}^{n}(1) \}$

Cross multiplication of equation (2)

$\sf:\longmapsto{ \frac{x}{a} - \frac{y}{b} = 4 } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{xb - ay = 4} \: \: \: \: \: \: \: ... \{eq {}^{n} \: (2) \}$

Add both equation

$\sf:\longmapsto{2xb + ay + (xb - ay) = 2 + 4} \\ \\ \sf:\longmapsto{2xb + xb + ay - ay = 6} \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{3xb = 6} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{xb = \cancel \frac{6}{3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{ \boxed{ \sf \red{xb = 2}}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Substitute xb = 2 in eqⁿ (1)

$\sf:\longmapsto{2(2) + ay = 2} \\ \\ \sf:\longmapsto{4 + ay = 2} \: \: \: \: \\ \\ \sf:\longmapsto{ay = 2 - 4} \: \: \: \\ \\ \sf:\longmapsto{ \boxed{ \sf{ \red{ay = - 2}}}} \: \: \: \:$

Answer :

• xb = 2
• ay = -2

Verification :-

Substitute xb = 2 and ay = -2 in eqⁿ (2)

$\sf:\longmapsto{2 - ( - 2) = 4} \\ \\ \sf:\longmapsto{2 + 2 = 4} \: \: \: \: \: \: \\ \\ \sf:\longmapsto{ {\underline{\underline{ \red{4 = 4}}}}} \: \: \: \: \: \: \: \: \: \: \: \: \:$

Hence Verified !

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