1.

Show that the system of equations ` 2x + 5y = 17, 5x + 3y = 14` has a unique solution. Find the solution.

Answer» The given system of equation is
` 2x + 5y - 17 = 0 , 5x + 3y - 14 = 0 `
These equations are of the form
`a_1 x + b_1 y + c _1= 0, a _2 x + b_2 y + c_2 =0 `,
where ` a_ 1 = 2, b_ 1 = 5, c_ 1 = - 17 and a_ 2 = 5, b_2 = 3, c_ 2 = -14`
` therefore (a _1 )/(a_ 2) = ( 2)/(5), (b_ 1 )/(b_ 2 ) = ( 5) /(3), (c _ 1 )/(c _ 2 ) = (-17)/(-14) = ( 17)/(14)`
Thus, ` (a_ 1 ) /(a _ 2 ) ne ( b_ 1 )/( b_ 2 ) `
Hence, the given system of equations has a unique solution.
By cross multiplication, we have
` (##RSA_MATH_X_C03_SLV_042_S01.png" width="80%">
` therefore (x) /(( - 70 + 51)) = ( y ) /(( - 85 + 28)) = (1)/(( 6- 25 ))`
` rArr (x)/(-19) = ( y ) /( -57) = (1 ) /(-19) `
` rArr x = (-19)/(-19) = 1 and y = (-57)/(-19) = 3` .
Hence, x= 1 and y = 3 is the required solution.


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