Solve the system of linear equations:x + y + z = 63x – 2y – z = 42x + 3y – 2z = 2

Sure, let's solve this system of equations step by step.

The system of equations is:

1. x + y + z = 6
2. 3x – 2y – z = 4
3. 2x + 3y – 2z = 2

Step 1: Let's subtract the first equation from the second and third equations to eliminate x:

Subtracting the first equation from the second gives: 2x - 3y = -2 Subtracting the first equation from the third gives: x + 2y - 3z = -4

So now we have a new system of equations:

1. x + y + z = 6
2. 2x - 3y = -2
3. x + 2y - 3z = -4

Step 2: Let's subtract the third equation from the second to eliminate y:

Subtracting the third equation from the second gives: x - 5y = 2

So now we have a new system of equations:

1. x + y + z = 6
2. x - 5y = 2
3. x + 2y - 3z = -4

Step 3: Let's solve the second equation for x:

x = 5y + 2

Step 4: Substitute x = 5y + 2 into the first and third equations:

Substituting into the first equation gives: 5y + 2 + y + z = 6, which simplifies to 6y + z = 4 Substituting into the third equation gives: 5y + 2 + 2y - 3z = -4, which simplifies to 7y - 3z = -6

So now we have a new system of equations:

1. 6y + z = 4
2. 7y - 3z = -6

Step 5: Multiply the first equation by 3 and the second equation by 1 to make the coefficients of z the same:

Multiplying the first equation by 3 gives: 18y + 3z = 12 Multiplying the second equation by 1 gives: 7y - 3z = -6

Step 6: Add the two equations together to eliminate z:

Adding the two equations gives: 25y = 6

Step 7: Solve for y:

y = 6 / 25

Step 8: Substitute y = 6 / 25 into the first equation to find z:

6*(6 / 25) + z = 4, which simplifies to z = 4 - 36 / 25 = 64 / 25

Step 9: Substitute y = 6 / 25 and z = 64 / 25 into the original first equation to find x:

x + 6 / 25 + 64 / 25 = 6, which simplifies to x = 6 - 70 / 25 = 80 / 25

So the solution to the system of equations is x = 80 / 25, y = 6 / 25, and z = 64 / 25.