Hello!
The answers are:
- A and B have different solutions.
- System C simplifies to 2x - 3y = 4 and 4x - y = 18 by dividing the second equation by three.
- Systems A and C have the same solution.
Why?
Let's solve each equation system in order to be able to choose the correct statements:
System A
[tex]2x-3y=4\\4x-y=18[/tex]
For the second equation we have:
[tex]y=4x-18[/tex]
Then, substituting y into the first equation:
[tex]2x-3(4x-18)=4\\2x-12x+54=4\\-10x=4-54\\-10x=-50\\x=5[/tex]
Then, substituting x into the second equation:
[tex]y=4(5)-18=2[/tex]
So, for the system A, we have:
x= 5
y= 2
System B
[tex]3x-4y=5\\y=5x+3[/tex]
Substituting y into the first equation we have:
[tex]3x-4(5x+3)=5\\3x-20x-12=5\\-17x-12=5\\-17x=12+5\\-17x=17\\x=-1[/tex]
Substituting x into the second equation, we have:
[tex]y=5(-1)+3=-2[/tex]
So, for the system B, we have:
x= -1
y= -2
System C
[tex]2x-3y=4\\12x-3y=54[/tex]
Dividing the second equation by three, we have:
[tex]4x-y=18\\y=4x-18[/tex]
Substituting y into the first equation, we have:
[tex]2x-3(4x-18)=4\\2x-12x+54=4\\-10x=4-54\\-10x=-50\\x=5[/tex]
Substituting x into the second equation, we have:
[tex]y=4(5)-18=2[/tex]
So, for the system C, we have:
x=5
y=2
Conclusions
System A and B have different solutions, since:
System A: X=5 and Y=2
System B: X=-1 and Y=-2
System C simplifies to 2x - 3y = 4 and 4x - y = 18 by dividing the second equation by three, since:
We did the simplification when we were solving the System C:
3(4x-y-18)=12x-3y-54=0
12x-3y=54
System A and C have the same solution, since:
System A: X=5 and Y=2
System C: X=5 and Y=2
Have a nice day!
