Answer:
The answer is the option C
\(3x-y=14\)
\(-9x+3y=-42\)
Step-by-step explanation:
case A) we have
\(-6x+3y=18\) ------> equation A
\(4x-3y=6\) ------> equation B
Adds equation A and equation B
\(-6x+3y=18 \\4x-3y=6\\--------- \\-6x+4x=18+6 \\-2x=24 \\ x=-12\)
Find the value of y
substitute in the equation A the value of x
\(-6(-12)+3y=18\)
\(3y=18-72\)
\(y=-18\)
therefore
the system case A) has one solution
case B) we have
\(2x+4y=24\) ------> equation A
\(6x+12y=36\) ------> equation B
Multiply by \(3\) equation A
\(3(2x+4y)=3*24\)
\(6x+12y=72\)
so
Equation A and equation B are parallel lines
therefore
the system case B) has no solution
case C) we have
\(3x-y=14\) ------> equation A
\(-9x+3y=-42\) ------> equation B
Multiply by \(3\) equation A
\(3(3x-y)=3*14\) -------> \(-9x+3y=-42\)
Equation A and equation B are the same line
therefore
the system case C) has infinitely solutions
case D) we have
\(5x+2y=13\) ------> equation A
\(-x+4y=-6\) ------> equation B
Multiply by \(5\) equation B
\(5(-x+4y)=-6*5\)
\(-5x+20y=-30\) -------> equation C
Adds equation A and equation C
\(5x+2y=13\\-5x+20y=-30\\------------\\2y+20y=13-30 \\22y=-17 \\y=-0.77\)
Substitute the value of y in the equation B
\(-x+4(-0.77)=-6\)
\(-x=-6+3.09\)
\(x=2.91\)
the system case D) has one solution
