x=-1, y=2, z=3
Step-by-step explanation:
First at all, you can divide these equations into two group. It is easier to you.
4x-2y+3z=1
3x+y+2z=5
x+3y-4z=-7
3x+y+2z=5
Then, you can simplify these group of equation that you divided.
So, we have to simplify these step by step.
First, we can simplify the first group of equation.
4x-2y+3z=1
3x+y+2z=5
We have to find which variable you want to eliminate. I choose y.
You can multiply the second equation of first group by 2 and eliminate them.
4x-2y+3z=1
6x+2y+4z=10
As the variable of y is same and we can eliminate y by addition.
4x-2y+3z=1
6x+2y+4z=10
become
10x+0+7z=11
10x+7z=11
10x+7z=11 as the third equation.
Next, simplify the second group of equation.
x+3y-4z=-7
3x+y+2z=5
Same way as the previous one. We need to find which variable that you want to eliminate. I choose y.
We need to balance for the first equation x+3y-4z=-7 at least one variable.
We need to multiply this equation by -3 below to get the same value of y to eliminate the above equation.
3x+y+2z=5 (multiply by -3)
-9x-3y-6z=-15
x+3y-4z= -7
-9x-3y-6z= -15
Here we go! Now these group has same value and variable to eliminate.
We can eliminate these equations by addition method.
x+3y-4z= -7
-9x-3y-6z= -15
-8x+0+10z= -22
-8x+10z= -22
Furthermore, we have to combine these equations that we have simplified from the previous groups.
10x+7z=11
-8x+10z= -22
10x+7z=11 and -8x+10z= -22 as a new group.
Finally, it is time to solve these equations!!
10x+7z=11
-8x+10z= -22
First equation of the above group, we can multiply by 10. Like this:
100x+70z=110
Second equation of the above group, we can multiply by 7. Like this:
-56x+70z=-154
So, we can see there are same value from these equation which are +70z and -70z.
Now, we can eliminate this variable by sum these equations.
100x+70z=110
-56x+70z=-154
44x= -44
x= -1
When we get x= -1 , we can use this value to any equation that you want. You can get the same answer as well.
When x= -1,
substitute into 100x+70z=110
100(-1)+70z=110
-100+70z=110
70z=110+100
70z=210
z=3
When z=3 and x= -1, we can substitute into x+3y-4z= -7
-1+3y-4(3)= -7
-1+3y-12= -7
3y=6
y=2
So, the answer is x= -1 , y=2 , z=3
I hope you will understand my solution and explanation. Bye!!
