Question

what is the solution to the following systems of equations ? x-3y+2z=11 -x+4y+3z=5 2x-2y-4z=2

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the system of equations:

Equation 1: x - 3y + 2z = 11
Equation 2: -x + 4y + 3z = 5
Equation 3: 2x - 2y - 4z = 2

First, we'll use the method of elimination to eliminate variables one by one.

Multiplying Equation 1 by 2 gives us:
2x - 6y + 4z = 22 -> Equation 4

Adding Equation 3 to Equation 4 eliminates the x variable:
(2x - 2y - 4z) + (2x - 6y + 4z) = 2 + 22
4x - 8y = 24 -> Equation 5

Next, multiplying Equation 3 by 3 gives us:
6x - 6y - 12z = 6 -> Equation 6

Adding Equation 5 to Equation 6 eliminates the y variable:
(4x - 8y) + (6x - 6y - 12z) = 24 + 6
10x - 14z = 30 -> Equation 7

Multiplying Equation 2 by 2 gives us:
-2x + 8y + 6z = 10 -> Equation 8

Adding Equation 1 to Equation 8 eliminates the y variable:
(x - 3y + 2z) + (-2x + 8y + 6z) = 11 + 10
- x + 8z = 21 -> Equation 9

Simplifying Equation 7 gives us:
5x - 7z = 15 -> Equation 10

Now we have a system of two equations to solve simultaneously:

Equation 10: 5x - 7z = 15
Equation 9: -x + 8z = 21

Multiplying Equation 9 by 5 and Equation 10 by -1 gives us:
-5x + 40z = 105 -> Equation 11
-5x + 7z = -15 -> Equation 12

Adding Equation 11 to Equation 12 eliminates the x variable:
(-5x + 40z) + (-5x + 7z) = 105 - 15
47z = 90

Solving for z gives us:
z = 90/47

Substituting z back into Equation 10:
5x - 7(90/47) = 15

Simplifying:
5x - (630/47) = 15
5x = 15 + (630/47)
5x = 705/47
x = (705/47) / 5
x = 705/235
x = 3

Substituting the values of x and z into Equation 9:
-3 + 8(90/47) = 21

Simplifying:
3 - (720/47) = 21
- (720/47) = 21 - 3
- (720/47) = 18
720/47 = -18

Therefore, the solution to the system of equations is:
x = 3
y = -18
z = 90/47

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the solution to the given system of equations, we can use the method of elimination or substitution. Let's use the method of elimination:

Step 1: Multiply Equation 1 by 2 and Equation 2 by -1 to eliminate x:
2(x - 3y + 2z) = 2(11) ⟹ 2x - 6y + 4z = 22
-1(-x + 4y + 3z) = -1(5) ⟹ x - 4y - 3z = -5

The new system of equations becomes:
2x - 6y + 4z = 22 ...(Equation 3)
x - 4y - 3z = -5 ...(Equation 4)
2x - 2y - 4z = 2 ...(Equation 5)

Step 2: Multiply Equation 4 by 2 and Equation 5 by -1 to eliminate x:
2(x - 4y - 3z) = 2(-5) ⟹ 2x - 8y - 6z = -10
-1(2x - 2y - 4z) = -1(2) ⟹ -2x + 2y + 4z = -2

The new system of equations becomes:
2x - 6y + 4z = 22 ...(Equation 6)
-2x + 2y + 4z = -2 ...(Equation 7)
2x - 8y - 6z = -10 ...(Equation 8)

Step 3: Add Equation 6 and Equation 7:
(2x - 6y + 4z) + (-2x + 2y + 4z) = 22 + (-2)
-4y + 8z = 20 ...(Equation 9)

Step 4: Multiply Equation 9 by 2:
2(-4y + 8z) = 2(20)
-8y + 16z = 40 ...(Equation 10)

Step 5: Add Equation 8 and Equation 10:
(2x - 8y - 6z) + (-8y + 16z) = -10 + 40
2x - 16y + 10z = 30 ...(Equation 11)

Step 6: Multiply Equation 11 by 2:
2(2x - 16y + 10z) = 2(30)
4x - 32y + 20z = 60 ...(Equation 12)

Step 7: Subtract Equation 12 from Equation 6:
(2x - 6y + 4z) - (4x - 32y + 20z) = 22 - 60
-2x + 26y - 16z = -38 ...(Equation 13)

Step 8: Multiply Equation 13 by 2:
2(-2x + 26y - 16z) = 2(-38)
-4x + 52y - 32z = -76 ...(Equation 14)

Step 9: Add Equation 12 and Equation 14:
(4x - 32y + 20z) + (-4x + 52y - 32z) = 60 + (-76)
20y - 12z = -16 ...(Equation 15)

Step 10: Multiply Equation 15 by 5:
5(20y - 12z) = 5(-16)
100y - 60z = -80 ...(Equation 16)

Step 11: Add Equation 16 and Equation 10:
(100y - 60z) + (-8y + 16z) = -80 + 40
92y - 44z = -40 ...(Equation 17)

Step 12: Divide Equation 17 by 4:
(92y - 44z) / 4 = -40 / 4
23y - 11z = -10 ...(Equation 18)

Step 13: Multiply Equation 18 by 2:
2(23y - 11z) = 2(-10)
46y - 22z = -20 ...(Equation 19)

Step 14: Add Equation 19 and Equation 9:
(46y - 22z) + (-4y + 8z) = -20 + 20
42y - 14z = 0 ...(Equation 20)

Step 15: Multiply Equation 20 by 3:
3(42y - 14z) = 3(0)
126y - 42z = 0 ...(Equation 21)

Step 16: Divide Equation 21 by 42:
(126y - 42z) / 42 = 0 / 42
3y - z = 0 ...(Equation 22)

Step 17: Rearrange Equation 22:
z = 3y

Now, we have obtained the relationship between z and y. Let's find the relationship between x, y, and z by substituting z = 3y back into any two of the three original equations.

Using Equation 4: x - 4y - 3z = -5
x - 4y - 3(3y) = -5
x - 4y - 9y = -5
x - 13y = -5 ...(Equation 23)

Using Equation 5: 2x - 2y - 4z = 2
2x - 2y - 4(3y) = 2
2x - 2y - 12y = 2
2x - 14y = 2 ...(Equation 24)

Now, we have obtained the relationship between x, y, and z in terms of two equations Equation 23 and Equation 24.

The solution to the given system of equations is:
x = 148y - 9
y = y
z = 3y

Answer 3

[AI-generated answer — Explain Bot]

To find the solution to the given system of equations, we can use the method of Gaussian elimination or matrix operations. Let's use the method of Gaussian elimination to solve the equations step-by-step:

Step 1: Write down the augmented matrix for the given system of equations:
```
[ 1 -3 2 | 11 ]
[ -1 4 3 | 5 ]
[ 2 -2 -4 | 2 ]
```

Step 2: Perform row operations to simplify the matrix. The goal is to get the matrix into row-echelon form or reduced row-echelon form. Here are the operations we'll perform:

- Add Row 1 to Row 2:
```
[ 1 -3 2 | 11 ]
[ 0 1 5 | 16 ]
[ 2 -2 -4 | 2 ]
```

- Subtract 2 times Row 1 from Row 3:
```
[ 1 -3 2 | 11 ]
[ 0 1 5 | 16 ]
[ 0 4 -8 | -20 ]
```

- Subtract 4 times Row 2 from Row 3:
```
[ 1 -3 2 | 11 ]
[ 0 1 5 | 16 ]
[ 0 0 -28 | -84 ]
```

- Divide Row 3 by -28:
```
[ 1 -3 2 | 11 ]
[ 0 1 5 | 16 ]
[ 0 0 1 | 3 ]
```

- Add 3 times Row 3 to Row 2:
```
[ 1 -3 2 | 11 ]
[ 0 1 0 | 25 ]
[ 0 0 1 | 3 ]
```

- Add 2 times Row 3 to Row 1:
```
[ 1 -3 0 | 17 ]
[ 0 1 0 | 25 ]
[ 0 0 1 | 3 ]
```

- Add 3 times Row 2 to Row 1:
```
[ 1 0 0 | 92 ]
[ 0 1 0 | 25 ]
[ 0 0 1 | 3 ]
```

Step 3: Read off the solutions for x, y, and z from the row-echelon form of the augmented matrix. From the augmented matrix obtained:
x = 92
y = 25
z = 3

Therefore, the solution to the given system of equations is x = 92, y = 25, z = 3.