How do you tell whether the system has one solution, infinitely many solutions, or no solution: #10y=-7x+18, 3.5x+5y=9#?

1 Answer
Jul 31, 2018

# 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."#

Explanation:

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."#

.....................................................................................................................
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."#
...........................................................................................................

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."#