Question

Solve the following systems of equations:
23x - 29y = 98
29x - 23y = 110

Answer 1

Concept :

When the coefficients of x and y in one equation interchanged in the other:

Step 1. Add the given equations and from it obtain an equation of the form x+ y = a.

Step 2. Subtract the given equations and from it obtain an equation of the form x -y = b.

Now , x + y = a and x - y = b can be solved easily.

SOLUTION:

23x - 29y = 98 …………...( 1 )

29x - 23y = 110 …………...( 2 )

On Adding Equation 1 and 2 :  

23x - 29y = 98  

29x - 23y = 110

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52x - 52y = 208

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52 (x - y) = 208

x - y = 208/52

x - y = 4 ……………….(3)

 

On Subtracting equation 1 and 2 :

23x - 29y = 98  

29x - 23y = 110

(-)    (+)     (-)

--------------------------

-6x - 6y = - 12

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-6(x + y) = - 12

X + y  = - 12/-6

x + y = 2…………...( 4 )

On adding equation 3 and 4

x + y = 2

x -  y = 4          [By elimination method]

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2x = 6

x = 6/2

x = 3

On substituting x = 3 in eq 3

x + y = 2

3 + y = 2

y = 2 - 3

y = - 1

Hence ,the solution of the given system of equation is x =  3 and y = -1.

Hope this answer will help you…

 

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Answer 2

$\bf \huge{ \red{Given}:-}$

$\bf \red{The \: given \: system \: of \: equation \: is}$

$\: \: \: \: \: \red→ 23x - 29y = 98 \: \: \: \: ...(i)$

$\: \: \: \: \: \: \: \: \: \bf → \: 29x - 23y = 110 \: \: \: ...(ii)$

Adding equation (i) and equation (ii), we

$\bf \: 23x \: + 29x - 29y = 98 + 110$

$\bf \:→52x - 52y = 208$

$\bf \: →52x - 52 = 208$

$\bf \: →52(x - y) = 208$

$\bf \: →x - y = \frac{208}{52} = 4$

$\bf \: →x - y - 4 \: \: \: ...(iii)$

Subtracting equation (i) by equation (ii), we get

$\bf \: 29x - 23x - 23y + 29y = 110 - 98$

$\bf \: →6x + 6y = 12$

$\bf \: →6(x + y) = 12$

$\bf \: →x + y = \frac{12}{6} = 2$

$\bf \: →x + y = 2 \: \: \: ...(iv)$

Adding equation (iii) and equation (iv), we get

$\bf \: 2x = 2 + 4 = 6$

$\bf \: →x = \frac{6}{2} = 3$

Putting x = 3 in equation (iv) , we get

$\bf \: 3 + y = 2$

$\bf \: →y = 2 - 3 = - 1$

Hence, solution of the given system of equation is x =3, y =-1