Question

find the value of k for which thr syste of equation is 2x+3y=7and ​

Answer 1

The given system of equations:

2x + 3y = 7,

⇒ 2x + 3y - 7 = 0 ….(i)

And, (k – 1)x + (k + 2)y = 3k

⇒(k – 1)x + (k + 2)y - 3k = 0 …(ii)

These equations are of the following form:

a1x+b1y+c1=0,a2x+b2y+c2=0

where,

a1=2,b1=3,c1=-7anda2=(k–1),b2=(k+2),c2=-3k

For an infinite number of solutions, we must have:

a1a2=b1b2=c1c2

2(c−1)=3(k+2)=−7−3k

⇒2(k−1)=3(k+2)=73k

Now, we have the following three cases:

Case I:

2(k−1)=3k+2

⇒ 2(k + 2) = 3(k – 1) ⇒ 2k + 4 = 3k – 3 ⇒ k = 7

Case II:3(k+2)=73k

⇒ 7(k + 2) = 9k ⇒ 7k + 14 = 9k ⇒ 2k = 14 ⇒ k = 7

Case III:

2(k−1)=73k

⇒ 7k – 7 = 6k ⇒ k = 7

Hence, the given system of equations has an infinite number of solutions when k is equal to 7.

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