We are given the following system of linear equations:

1. $3x + 2y + z = 2$
2. $4x + 2y + 2z = 8$
3. $2x - 2y + 2z = 8$

We will solve this system using the elimination method. The goal is to eliminate one of the variables by adding or subtracting the equations.

Step 1: Eliminate one variable

Let's begin by eliminating $y$ from the first two equations. To do this, we need to manipulate the equations such that the coefficients of $y$ are the same.

• From equation (1): $3x + 2y + z = 2$
• From equation (2): $4x + 2y + 2z = 8$

Subtract equation (1) from equation (2) to eliminate $y$:

$(4x + 2y + 2z) - (3x + 2y + z) = 8 - 2$

This simplifies to:

$4x + 2y + 2z - 3x - 2y - z = 6$

$x + z = 6 \quad \text{(Equation 4)}$

Step 2: Eliminate $y$ again using equations (1) and (3)

Now let's eliminate $y$ from equations (1) and (3):

• From equation (1): $3x + 2y + z = 2$
• From equation (3): $2x - 2y + 2z = 8$

Multiply equation (1) by 1 and equation (3) by 1 (to keep them in their current form) and add them to eliminate $y$:

$(3x + 2y + z) + (2x - 2y + 2z) = 2 + 8$

This simplifies to:

$3x + 2y + z + 2x - 2y + 2z = 10$

$5x + 3z = 10 \quad \text{(Equation 5)}$

Step 3: Solve the system of two equations with two variables

We now have two equations with two variables, $x$ and $z$:

• $x + z = 6$ (Equation 4)
• $5x + 3z = 10$ (Equation 5)

Step 3.1: Solve for $x$ in terms of $z$ from equation (4):

$x = 6 - z$

Step 3.2: Substitute this expression for $x$ into equation (5):

$5(6 - z) + 3z = 10$

$30 - 5z + 3z = 10$

$30 - 2z = 10$

$-2z = 10 - 30$

$-2z = -20$

$z = 10$

Step 3.3: Substitute $z = 10$ into the expression for $x$:

$x = 6 - 10 = -4$

Step 4: Solve for $y$

Now that we have values for $x$ and $z$, substitute $x = -4$ and $z = 10$ into one of the original equations, for example, equation (1):

$3x + 2y + z = 2$

$3(-4) + 2y + 10 = 2$

$-12 + 2y + 10 = 2$

$-2 + 2y = 2$

$2y = 4$

$y = 2$

Final Solution:

The solution to the system of equations is:

$x = -4, \quad y = 2, \quad z = 10$

Would you like a more detailed breakdown of any step, or is there another concept you'd like to explore?

Additional Related Questions:

1. How can you check the solution to a system of equations?
2. What happens if one of the equations in the system is dependent on the others?
3. Can you solve a system of three linear equations with a different method, such as substitution?
4. What would happen if the system had no solution? How can you detect that?
5. How does the elimination method compare to the substitution method in terms of efficiency?

Tip:

When solving systems of equations, always check if the equations can be simplified first (e.g., by multiplying or dividing to make the coefficients easier to work with) before proceeding with elimination or substitution.