Find the value of F for which each of the following system of equations having infinitely many solution: $2x–3y=7$, $(F+2)x–(2F+1)y=3(2F-1)$

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Find the value of F for which each of the following system of equations having infinitely many solution: $2x-3y=7$ , $(F+2)x-(2F+1)y=3(2F-1)$

Find the value of F for which each of the following system of equations having infinitely many solution: $2x-3y=7$ , $(F+2)x-(2F+1)y=3(2F-1)$

The given system of equations is:

$2x-3y-7=0$

$(F+2)x-(2F+1)y-3(2F-1)=0$

The above equations are of the form

${{a}_{1}}x+{{b}_{1}}y-{{c}_{1}}=0$

${{a}_{2}}x+{{b}_{2}}y-{{c}_{2}}=0$

Here, ${{a}_{1}}=2,{{b}_{1}}=-3,{{c}_{1}}=-7$

${{a}_{2}}=\left( F+2 \right),{{b}_{2}}=-\left( 2F+1 \right),{{c}_{2}}=-3\left( 2F-1 \right)1$

So according to the question,

For unique solution, the condition is

${{a}_{1}}/{{a}_{2}}={{b}_{1}}/{{b}_{2}}={{c}_{1}}/{{c}_{2}}$

$2/\left( F+2 \right)=-3/-\left( 2F+1 \right)=-7/-3\left( 2F-1 \right)$

$2/\left( F+2 \right)=-3/-2\left( F+1 \right)$ and $-3/-\left( 2F+1 \right)=-7/-3\left( 2F-1 \right)$

$\Rightarrow 2/\left( 2F+1 \right)=3\left( F+2 \right)$ and $3\times 3\left( 2F-1 \right)=7\left( 2F+1 \right)$

$\Rightarrow 4F+2=3F+6$ and $18F-9=14F+7$

$\Rightarrow F=4$ and $4F=16\Rightarrow F=4$

Therefore, the given system of equations will have infinitely many solutions, if $F=4$ .

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