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