we know that
If a system has exactly one solution, it is a consistent independent system.
If a system has an infinite number of solutions, it is a consistent dependent system .
If a system has no solution, it is said to be inconsistent
We're going to solve each system to determine its classification
case A)
[tex]4x-2y=10[/tex] ------> equation A
[tex]2x-y=5[/tex] ------> equation B
Multiply equation B by [tex]2[/tex] both sides
[tex]2*(2x-y)=2*5[/tex] -----> [tex]4x-2y=10[/tex]
Equation A and equation B are the same line
Therefore
The system has an infinite number of solutions
The answer case A) is
The system is a consistent dependent system (coincident)
case B)
[tex]y=2x-3[/tex] ------> equation A
[tex]-2x+y=-5[/tex] ------> equation B
Isolate the variable y in the equation B
[tex]y=2x-5[/tex]
Both lines are parallel lines, because has the same slope
Therefore
The system has no solution
the answer case B) is
The system is inconsistent
case C)
[tex]2x-5y=14[/tex] ------> equation A
[tex]3x+4y=10[/tex] ------> equation B
Multiply equation A by [tex]4[/tex] and equation B by [tex]5[/tex] both sides
[tex]4*(2x-5y)=4*14[/tex] -----> [tex]8x-20y=56[/tex]
[tex]5*(3x+4y)=5*10[/tex] -----> [tex]15x+20y=50[/tex]
Adds the equations
[tex]8x-20y=56 \\15x+20y=50\\-------\\8x+15x=56+50 \\23x=106 \\x=4.61[/tex]
Find the value of y
[tex]2*4.6-5y=14[/tex]
[tex]y=-0.96[/tex]
therefore
The system has one solution
The answer case C) is
The system is a consistent independent system
