Find the value of F for which each of the following system of equations having infinitely many solution$2a+3y–5=0$, $6a+Fy–15=0$

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Find the value of F for which each of the following system of equations having infinitely many solution $2a+3y-5=0$ , $6a+Fy-15=0$

Find the value of F for which each of the following system of equations having infinitely many solution $2a+3y-5=0$ , $6a+Fy-15=0$

The Given equations are:

$2a+3y-5=0$

$6a+Fy-15=0$

Above equations are of the form

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

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

Now, ${{a}_{1}}=2,{{b}_{1}}=3,{{c}_{1}}=-5$

${{a}_{2}}=6,{{b}_{2}}=F,{{c}_{2}}=-15$
According to the question,

For unique solution, the condition is

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

$2/6=3/F$

$\Rightarrow F=9$

Therefore, given equations will have infinitely many solutions, if $F=9$ .

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