Question

What is the solution of the system of equations?-3x-4y-3z=-7

2x-6y+2z=3
5x-2y+5z=9

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the solution of the system of equations:

-3x - 4y - 3z = -7
2x - 6y + 2z = 3
5x - 2y + 5z = 9

One way to approach this is by using the method of elimination. We can eliminate one variable at a time by manipulating the equations. Let's start by eliminating z.

Multiply the first equation by 2 and the second equation by 3 to create the same coefficient for z:
-6x - 8y - 6z = -14
6x - 18y + 6z = 9

Add the two equations together to eliminate z:
-6x - 8y - 6z + 6x - 18y + 6z = -14 + 9
-26y = -5
y = 5/26

Substitute the value of y back into the first equation to solve for x:
-3x - 4(5/26) - 3z = -7
-3x - 20/26 - 3z = -7
-3x - 10/13 - 3z = -7
-3x - 10/13 - 3z = -7
-3x - 3z = -7 + 10/13
-3x - 3z = -91/13

Next, we need to eliminate z again. Multiply the second equation by 5 and the third equation by 2 to create the same coefficient for z:
10x - 30y + 10z = 15
10x - 4y + 10z = 18

Subtract the two equations to eliminate z:
10x - 30y + 10z - 10x + 4y - 10z = 15 - 18
-26y = -3
y = 3/26

Substitute the value of y back into the second equation to solve for z:
2x - 6(3/26) + 2z = 3
2x - 18/26 + 2z = 3
2x - 9/13 + 2z = 3
2x + 2z = 3 + 9/13
2x + 2z = 48/13

Now we have two equations with two variables:
-3x - 3z = -91/13
2x + 2z = 48/13

Multiply the second equation by -3 to create opposite coefficients for x:
-3(-3x - 3z) = -3(-91/13)
9x + 9z = 273/13

Add this equation to the first equation to eliminate x:
(-3x - 3z) + (9x + 9z) = -91/13 + 273/13
6z = 182/13
z = 182/78

Now substitute the value of z back into the second equation to solve for x:
2x + 2(182/78) = 48/13
2x + 364/78 = 48/13
2x + 182/39 = 48/13
25x + 182 = 192
25x = 10
x = 10/25
x = 2/5

Therefore, the solution to the system of equations is x = 2/5, y = 3/26, and z = 182/78.

Answer 2

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

To find the solution of the given system of equations, we will use the method of elimination.

Step 1: Multiply the second equation by 3 and the third equation by -1 to eliminate the variable "x".

3*(2x-6y+2z) = 3*3 => 6x - 18y + 6z = 9
-1*(5x-2y+5z) = -1*9 => -5x + 2y - 5z = -9

The system becomes:
-3x - 4y - 3z = -7
6x - 18y + 6z = 9
-5x + 2y - 5z = -9

Step 2: Add the first and second equations to eliminate the "x" variable.

-3x - 4y - 3z + (6x - 18y + 6z) = -7 + 9
3x - 22y + 3z = 2

The system becomes:
3x - 22y + 3z = 2
-5x + 2y - 5z = -9

Step 3: Multiply the first equation by 2 and the second equation by 3 to eliminate the variable "y".

2*(3x - 22y + 3z) = 2*2 => 6x - 44y + 6z = 4
3*(-5x + 2y - 5z) = 3*(-9) => -15x + 6y - 15z = -27

The system becomes:
6x - 44y + 6z = 4
-15x + 6y - 15z = -27

Step 4: Add the first and second equations to eliminate the "y" variable.

(6x - 44y + 6z) + (-15x + 6y - 15z) = 4 + (-27)
-9x - 38z = -23

The system becomes:
-9x - 38z = -23

Step 5: Solve for "z".

-9x - 38z = -23
Let's express "x" in terms of "z" by dividing the equation by -9:
x = (38z - 23) / 9

Therefore, the solution of the given system of equations is:
x = (38z - 23) / 9
y = (13z + 5) / 6
z = z

Note: The solution is given in terms of "z" because the system of equations is consistent and linearly independent, meaning there is no unique solution. The values of "x", "y", and "z" can vary depending on the chosen value of "z".

Answer 3

[AI-generated answer — Explain Bot]

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

Step 1: Rearrange the equations to align the variables:
The original system of equations is:
-3x - 4y - 3z = -7 (Equation 1)
2x - 6y + 2z = 3 (Equation 2)
5x - 2y + 5z = 9 (Equation 3)

Step 2: Multiply Equation 1 by 2 and Equation 2 by 3 to create opposite coefficients of x:
-6x - 8y - 6z = -14 (Equation 1 multiplied by 2)
6x - 18y + 6z = 9 (Equation 2 multiplied by 3)
5x - 2y + 5z = 9 (Equation 3)

Step 3: Add Equation 1 multiplied by 2 to Equation 2 to eliminate the x term:
(-6x - 8y - 6z) + (6x - 18y + 6z) = -14 + 9
-26y = -5
Divide by -26:
y = -5/-26
y = 5/26

Step 4: Substitute the value of y in one of the original equations to solve for x and z.
Let's substitute y = 5/26 in Equation 1:
-3x - 4(5/26) - 3z = -7
-3x - 20/26 - 3z = -7
Simplify the equation by multiplying each term by 26 to remove the fraction:
-78x - 80 - 78z = -182

Step 5: Simplify and rearrange the equation:
-78x - 78z = -182 + 80
-78x - 78z = -102
Divide the equation by -78 to isolate x and z:
x + z = 102/78
x + z = 17/13

Step 6: Now, let's write the equations for x, y, and z using the values we found:
x = 17/13
y = 5/26
z = 17/13 - x

So, the solution to the system of equations is:
x = 17/13
y = 5/26
z = 17/13 - x