Find the value of F for which each of the following system of equations having infinitely many solution: a+(F+1)y=4, (F+1)a+9y=5F+2
Find the value of F for which each of the following system of equations having infinitely many solution: a+(F+1)y=4, (F+1)a+9y=5F+2

The given system of equations is:

a+(F+1)y-4=0

(F+1)a+9y-(5F+2)=0

The above equations are of the form

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

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

Here,{{a}_{1}}=1,{{b}_{1}}=\left( F+1 \right),{{c}_{1}}=-4

{{a}_{2}}=\left( F+1 \right),{{b}_{2}}=9,{{c}_{2}}=-\left( 5F+2 \right)

So according to the question,

For unique solution, the condition is

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

1/F+1=\left( F+1 \right)/9=-4/-\left( 5F+2 \right)

1/F+1=F+1/9And F+1/9=4/5F+2

\Rightarrow 9={{\left( F+1 \right)}^{2}}and \left( F+1 \right)\left( 5F+2 \right)=36

\Rightarrow 9={{F}^{2}}+2F+1And 5{{F}^{2}}+2F+5F+2=36

\Rightarrow {{F}^{2}}+2F-8=0And 5{{F}^{2}}+7F-34=0

\Rightarrow {{F}^{2}}+4F-2F-8=0and 5{{F}^{2}}+17F-10F-34=0

\Rightarrow F\left( F+4 \right)-2\left( F+4 \right)=0and \left( 5F+17 \right)-2\left( 5F+17 \right)=0

\Rightarrow \left( F+4 \right)\left( F-2 \right)=0And \left( 5F+17 \right)\left( F-2 \right)=0

\Rightarrow F=-4or F=2and F=-17/5or F=2

It’s seen that F=2 satisfies both the condition.

Thus, the given system of equations will have infinitely many solutions, if F=9.