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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. These equations are of the forms:  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:  a1/a2 ≠ b1/b2 , i.e., 2/1 ≠ 3/−2  Hence, the given system of equations has a unique solution.  Again, the given equations are:

2x + 3y – 18 = 0 …..(iii)

x – 2y – 2 = 0 …..(iv)

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

4x + 6y – 36 = 0 …….(v)

3x – 6y – 6 = 0 ……(vi)

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

7x = 42  ⇒x = 6

On substituting x = 6 in (iii),

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

⇒3y = (18 – 12) = 6

⇒y = 2

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

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