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# Show that the following systems of equations has a unique solution and solve it: x/3+y/2=3, x-2y=2

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This is the basic and exam oriented question from linear equations in two variables in which we have given two equations x/3+y/2=3, x-2y=2 and we have been asked to find the value of variables x and y from the given equations in such a way that it has a unique solution

Kindly give me a detailed solution of this question

RS Aggarwal, Class 10, chapter 3D, question no 3

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### 1 Answer

1. These equations are of the forms:Â  a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0

where, a1 = 2, b1= 3, c1 = -18 and a2 = 1, b2 = -2, c2 = -2

For a unique solution, we must have:Â  a1/a2 â‰  b1/b2 , i.e., 2/1 â‰  3/âˆ’2Â  Hence, the given system of equations has a unique solution.Â  Again, the given equations are:

2x + 3y â€“ 18 = 0 â€¦..(iii)

x â€“ 2y â€“ 2 = 0 â€¦..(iv)

On multiplying (i) by 2 and (ii) by 3, we get:

4x + 6y â€“ 36 = 0 â€¦â€¦.(v)

3x – 6y â€“ 6 = 0 â€¦â€¦(vi)

On adding (v) from (vi), we get:

7x = 42Â  â‡’x = 6

On substituting x = 6 in (iii),

we get: 2(6) + 3y = 18

â‡’3y = (18 – 12) = 6

â‡’y = 2

Hence, x = 6 and y = 2 is the required solution

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