{4х+2у+3z=9 3x-y-z=1 2x-5y+2z=-1

To solve the system of equations:

1. \( 4x + 2y + 3z = 9 \) (Equation 1)
2. \( 3x - y - z = 1 \) (Equation 2)
3. \( 2x - 5y + 2z = -1 \) (Equation 3)

We can use the method of substitution or elimination. Here, let's use elimination.

First, we'll organize the equations for easier manipulation:

Step 1: Simplifying Equations

We'll start with Equation 2, which is simpler, and try to express \( z \) in terms of \( x \) and \( y \).

From Equation 2, we can express \( z \):

\[ z = 3x - y - 1 \quad \text{(Equation 4)} \]

Step 2: Substitute \( z \) into other equations

Now we'll substitute Equation 4 into Equation 1 and Equation 3.

Substituting into Equation 1:

\[ 4x + 2y + 3(3x - y - 1) = 9 \]

Expanding this gives:

\[ 4x + 2y + 9x - 3y - 3 = 9 \]

Combine like terms:

\[ (4x + 9x) + (2y - 3y) - 3 = 9 \]

This simplifies to:

\[ 13x - y - 3 = 9 \] \[ 13x - y = 12 \quad \text{(Equation 5)} \]

Substituting into Equation 3:

Now we substitute Equation 4 into Equation 3:

\[ 2x - 5y + 2(3x - y - 1) = -1 \]

Expanding this gives:

\[ 2x - 5y + 6x - 2y - 2 = -1 \]

Combine like terms:

\[ (2x + 6x) + (-5y - 2y) - 2 = -1 \]

This simplifies to:

\[ 8x - 7y - 2 = -1 \] \[ 8x - 7y = 1 \quad \text{(Equation 6)} \]

Step 3: Solve the new system of equations

We now have a new system of two equations:

1. \( 13x - y = 12 \) (Equation 5)
2. \( 8x - 7y = 1 \) (Equation 6)

Let's solve Equation 5 for \( y \):

From \( 13x - y = 12 \):

\[ y = 13x - 12 \quad \text{(Equation 7)} \]

Step 4: Substitute \( y \) in Equation 6

Now, substitute Equation 7 into Equation 6:

\[ 8x - 7(13x - 12) = 1 \]

Expanding this gives:

\[ 8x - 91x + 84 = 1 \]

Combine like terms:

\[ -83x + 84 = 1 \] \[ -83x = 1 - 84 \] \[ -83x = -83 \] So,

\[ x = 1 \]

Step 5: Solve for \( y \) and \( z \)

Now that we have \( x \), we can find \( y \) using Equation 7:

\[ y = 13(1) - 12 = 1 \]

Next, use Equation 4 to find \( z \):

\[ z = 3(1) - 1 - 1 = 1 \]

Final Solution

So, the solution to the system of equations is:

\[ x = 1, \quad y = 1, \quad z = 1 \]

You can verify by substituting \( x, y, z \) back into the original equations to confirm they hold true.