We know that,
#"The systems of equations :"#
#a_1x+b_1y+c_1=0 , a_1^2+b_1^2!=0#
#a_2x+b_2y+c_2=0 , a_2^2+b_2^2!=0#
#"and let " a_1,b_1,c_1,a_2,b_2,c_2" are non zero ."#
#(i) a_1b_2-a_2b_1!=0 =>"one solution."#
#(ii) a_1b_2-a_2b_1=0#
but #b_1c_2-b_2c_1!=0 or c_1a_2-c_2a_1!=0=>"no solution"#
#(iii)a_1b_2-a_2b_1=b_1c_2-b_2c_1=c_1a_2-c_2a_1=0#
#=>"infinitely many solutions."#
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Here ,
#10y=-7x+18 and 3.5x+5y=9#
#:.7x+10y-18=0 to(1)and 3.5x+5y-9=0to(2)#
#=>a_1=7,b_1=10,c_1=-18 , a_2=3.5,b_2=5,c_2=-9#
So,
#a_1b_2-a_2b_1=(7xx5)-(3.5xx10)=35-35=0#
#b_1c_2-b_2c_1=(10xx(-9))-(5xx(-18))=-90+90=0#
#c_1a_2-c_2a_1=(-18xx3.5)-(-9xx7)=-63+63=0#
#i.e. color(red)(a_1b_2-a_2b_1=b_1c_2-b_2c_1=c_1a_2-c_2a_1=0#
#=>color(blue)("The system of equations have infinitely many solutions."#
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Note:
Multiplying eqn. #(2)# by #2#
#=>(3.5x+5y-9)xx2=0 xx 2#
#=>7x+10y-18=0to (2)#
#i.e. "both the eqns . are same."=>"infinitely many solutions."#
