Question

what is the solution to the following systems of equations ?

x-y +z=-1
x+y+3z=-3
2x-y+2z=0
show all steps

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve this system of equations, we can use the method of elimination or substitution. I will use the method of elimination.

Step 1: Multiply equation 1 by 2 and equation 3 by -1.
2(x - y + z) = 2(-1)
-1(2x - y + 2z) = -1(0)
This simplifies the equations to:
2x - 2y + 2z = -2
-2x + y - 2z = 0

Step 2: Add equation 2 and equation 3 to eliminate the x variable.
(x + y + 3z) + (-2x + y - 2z) = -3 + 0
This simplifies to:
-y + z = -3

Step 3: Multiply equation 2 by 2 and add it to equation 1 to eliminate the x variable.
2(x + y + 3z) + (x - y + z) = 2(-3) + (-1)
This simplifies to:
3x + 5z = -7

Step 4: Multiply equation 3 by 2 and add it to equation 1 to eliminate the x variable.
2(2x - y + 2z) + (x - y + z) = 2(0) + (-1)
This simplifies to:
5x + 4z = -2

Step 5: Now we have the simplified system of equations:
-y + z = -3
3x + 5z = -7
5x + 4z = -2

Step 6: Solve equation 1 for y:
y = z - 3

Step 7: Substitute y into equation 2 and equation 3:
3x + 5z = -7
5x + 4z = -2

Step 8: Solve equation 2 for x:
x = (-2 - 4z) / 5

Step 9: Substitute x and y into equation 1:
(-2 - 4z) / 5 - z + 3 = 0

Step 10: Solve equation 9 for z:
(-2 - 4z - 5z + 15) / 5 = 0
(-2 - 9z + 15) / 5 = 0
(-9z + 13) / 5 = 0
-9z + 13 = 0
-9z = -13
z = 13/9

Step 11: Substitute z back into equation 8 to find x:
x = (-2 - 4(13/9)) / 5
x = (-2 - 52/9) / 5
x = (-18/9 - 52/9) / 5
x = (-70/9) / 5
x = -(70/9) * (1/5)
x = -14/9

Step 12: Substitute z and x into equation 6 to find y:
y = z - 3
y = 13/9 - 3
y = 13/9 - 27/9
y = -14/9

Therefore, the solution to the system of equations is:
x = -14/9, y = -14/9, z = 13/9.

Answer 2

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

To solve the system of equations:

Step 1: Start by writing the system of equations:
Equation 1: x - y + z = -1
Equation 2: x + y + 3z = -3
Equation 3: 2x - y + 2z = 0

Step 2: Choose two pairs of equations and eliminate one variable:
Let's eliminate the y variable by adding Equation 1 and Equation 3 together.
Equation 1 + Equation 3: (x - y + z) + (2x - y + 2z) = (-1) + (0)

Simplifying the equation, we get:
3x - 2y + 3z = -1

Step 3: Now we have two simplified equations:
Equation 2: x + y + 3z = -3
Equation 4: 3x - 2y + 3z = -1

Step 4: To eliminate the y variable again, multiply Equation 2 by 2 and add it to Equation 4:
2 * Equation 2 + Equation 4: (2x + 2y + 6z) + (3x - 2y + 3z) = (-6) + (-1)

Simplifying the equation, we get:
5x + 9z = -7

Step 5: We now have two simplified equations:
Equation 5: 5x + 9z = -7
Equation 4: 3x - 2y + 3z = -1

Step 6: At this point, we can solve for x in Equation 5:
5x = -7 - 9z
Divide both sides by 5:
x = (-7 - 9z)/5

Step 7: Substitute the value of x in terms of z into Equation 4 to solve for y:
3((-7 - 9z)/5) - 2y + 3z = -1

Simplifying the equation, we get:
(-21 - 27z)/5 - 2y + 3z = -1

Step 8: We can multiply all terms by 5 to eliminate the fraction:
(-21 - 27z) - 10y + 15z = -5

Step 9: Combine like terms:
-12y - 12z = 16

Step 10: Divide both sides by -12:
y + z = -16/12
Simplify the right side:
y + z = -4/3

Step 11: We now have three equations:
Equation 5: 5x + 9z = -7
Equation 4: (-21 - 27z)/5 - 2y + 3z = -1
Equation 6: y + z = -4/3

Step 12: We can solve for y using Equation 6:
y = -4/3 - z

Step 13: Substitute the value of y in terms of z into Equation 4 to solve for x:
(-21 - 27z)/5 - 2((-4/3) - z) + 3z = -1

Simplifying the equation, we get:
(-21 - 27z)/5 + 8/3 + 2z + 3z = -1

Step 14: To eliminate the fractions, multiply all terms by 15:
3(-21 - 27z) + 40 + 30z + 45z = -15

Simplifying the equation, we get:
-63 - 81z + 40 + 75z = -15

Step 15: Combine like terms:
-6z = 8

Step 16: Divide both sides by -6:
z = -8/6
Simplify the right side:
z = -4/3

Step 17: Substitute the value of z into Equation 6 to solve for y:
y + (-4/3) = -4/3

Simplifying the equation, we get:
y = 0

Step 18: Substitute the values of y and z into Equation 5 to solve for x:
5x + 9(-4/3) = -7

Simplifying the equation, we get:
5x - 12 = -7

Step 19: Add 12 to both sides:
5x = 5

Step 20: Divide both sides by 5:
x = 1

The solution to the given system of equations is x = 1, y = 0, and z = -4/3.

Answer 3

[AI-generated answer — Explain Bot]

To find the solution to the system of equations, we will use the method of elimination or substitution. I will explain both methods so you can choose which one to apply.

1. Method of Elimination:
Step 1: Rearrange the equations to align the variables.
Equation 1: x - y + z = -1
Equation 2: x + y + 3z = -3
Equation 3: 2x - y + 2z = 0

Step 2: Multiply Equation 1 by 2 and Equation 2 by -1 to eliminate the x term.
2(x - y + z) = 2(-1) --> Equation 4: 2x - 2y + 2z = -2
-(x + y + 3z) = -1 --> Equation 5: -x - y - 3z = 3

Step 3: Add Equation 4 to Equation 5 to eliminate the x term.
(2x - 2y + 2z) + (-x - y - 3z) = -2 + 3
x - 4z = 1 --> Equation 6

Step 4: Subtract Equation 1 from Equation 3 to eliminate the x term.
(2x - y + 2z) - (x - y + z) = 0 - (-1)
x + z = 1 --> Equation 7

Step 5: Solve Equations 6 and 7 simultaneously.
Equation 6: x - 4z = 1 --> Equation 8 by multiplying the entire equation by -1: -x + 4z = -1
Equation 7: x + z = 1

Add Equation 8 to Equation 7:
(-x + 4z) + (x + z) = -1 + 1
5z = 0
z = 0

Substitute the value of z = 0 into Equation 7:
x + 0 = 1
x = 1

Substitute the values of x = 1 and z = 0 into Equation 1 or Equation 2:
x - y + z = -1 or x + y + 3z = -3
1 - y + 0 = -1 or 1 + y + 3(0) = -3
1 - y = -1 or 1 + y = -3
-y = -2 or y = -4
y = 2 or y = -4

Therefore, the solution to the given system of equations is x = 1, y = 2, and z = 0.

2. Method of Substitution:
In this method, we will solve one equation for one variable and substitute it into the other equations.

Step 1: Rearrange the equations to align the variables.
Equation 1: x - y + z = -1
Equation 2: x + y + 3z = -3
Equation 3: 2x - y + 2z = 0

Step 2: Solve Equation 1 for x:
x = -1 + y - z --> Equation 9

Step 3: Substitute Equation 9 into Equations 2 and 3:
Equation 2: (-1 + y - z) + y + 3z = -3 --> Equation 10
Equation 3: 2(-1 + y - z) - y + 2z = 0 --> Equation 11

Step 4: Simplify Equation 10 and Equation 11:
Equation 10: 2y + 2z = -2
Equation 11: 2y - z = 2

Step 5: Solve Equation 10 for y:
y = -z - 1 --> Equation 12

Step 6: Substitute Equation 12 into Equation 11:
2(-z - 1) - z = 2
-2z - 2 - z = 2
-3z = 4
z = -4/3

Step 7: Substitute the value of z = -4/3 into Equation 12:
y = -(-4/3) - 1
y = 4/3 - 1
y = 4/3 - 3/3
y = 1/3

Step 8: Substitute the values of y = 1/3 and z = -4/3 into Equation 9:
x = -1 + (1/3) - (-4/3)
x = -1 + 1/3 + 4/3
x = 3/3 + 5/3
x = 8/3
x = 2 2/3

Therefore, the solution to the given system of equations is x = 2 2/3, y = 1/3, and z = -4/3.

These are the step-by-step solutions using both the Method of Elimination and the Method of Substitution. Feel free to choose the method that you are more comfortable with or that suits your preferences.