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# Prove that There is a Value of C (≠ 0) for Which the System 6x + 3y = C - 3 12x + Cy = C Has Infinitely Many Solutions. Find this Value. - CBSE Class 10 - Mathematics

ConceptPair of Linear Equations in Two Variables

#### Question

Prove that there is a value of c (≠ 0) for which the system

6x + 3y = c - 3

12x + cy = c

has infinitely many solutions. Find this value.

#### Solution

The given system of equation may be written as

6x + 3y - (c - 3) = 0

12 - cy - c = 0

This is of the form

a_1x + b_1y + c_1 = 0

a_2x + b_2y + c_2 = 0

Where a_1 = 6, b_1 = 3 , c_1 = -(c - 3)

And a_2 = 12,b_2 = c, c_2 = -c

For infinitely many solutions, we must have

a_1/a_2 - b_1/b_2 = c_1/c_2

=> 6/12 = 13/c = (-(c - 3))/(-c)

=> 6/12 = 13/c = 3/c = (c- 3)/c

=> 6c = 12 xx 3 and 3 = (c - 3)

=> c = 36/6 and  c - 3 = 3

=> c = 6 and c = 6

Now

a_1/a_2 = 6/12 = 1/2

b_1/b_2 = 3/6 = 1/2

c_1/c_2 = (-(6-3))/(-6) = 1/2

:. a_1/a_2 = b_1/b_2 = c_1/c_2

Clearly, for this value of c, we have a_1/a_2 = b_1/b_2 = c_1/c_2

Hence, the given system of equations has infinitely many solutions if c = 6

Is there an error in this question or solution?

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Solution Prove that There is a Value of C (≠ 0) for Which the System 6x + 3y = C - 3 12x + Cy = C Has Infinitely Many Solutions. Find this Value. Concept: Pair of Linear Equations in Two Variables.
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