x-y=3
2x+3y+4z=17
y+2z=7

Answer

To solve the system of linear equations given by:

1. \( x - y = 3 \) (Equation 1)
2. \( 2x + 3y + 4z = 17 \) (Equation 2)
3. \( y + 2z = 7 \) (Equation 3)

we will follow these steps:

### Step 1: Eliminate \( y \) from Equations 1 and 3

From Equation 1, we can express \( y \) in terms of \( x \):

\[ y = x - 3 \]

Now, substitute this expression for \( y \) into Equation 3:

\[ (x - 3) + 2z = 7 \]

### Step 2: Simplify the equation

Now, simplify the equation:

\[ x - 3 + 2z = 7 \]

Adding 3 to both sides:

\[ x + 2z = 10 \quad \text{(Equation 4)} \]

### Step 3: Substitute \( y \) into Equation 2

Next, substitute \( y = x - 3 \) into Equation 2:

\[ 2x + 3(x - 3) + 4z = 17 \]

### Step 4: Expand and simplify

Expanding the equation gives:

\[ 2x + 3x - 9 + 4z = 17 \]

Combine like terms:

\[ 5x + 4z - 9 = 17 \]

Adding 9 to both sides:

\[ 5x + 4z = 26 \quad \text{(Equation 5)} \]

### Step 5: Solve for \( x \) and \( z \)

Now we have two equations (Equation 4 and Equation 5):

1. \( x + 2z = 10 \) (Equation 4)
2. \( 5x + 4z = 26 \) (Equation 5)

From Equation 4, we can express \( x \) in terms of \( z \):

\[ x = 10 - 2z \]

### Step 6: Substitute \( x \) into Equation 5

Substituting \( x \) into Equation 5:

\[ 5(10 - 2z) + 4z = 26 \]

### Step 7: Simplify and solve for \( z \)

Expanding gives:

\[ 50 - 10z + 4z = 26 \]

Combine like terms:

\[ 50 - 6z = 26 \]

Subtract 50 from both sides:

\[ -6z = 26 - 50 \]

\[ -6z = -24 \]

Dividing by -6:

\[ z = 4 \]

### Step 8: Substitute \( z \) back to find \( x \)

Now substitute \( z = 4 \) back into Equation 4 to find \( x \):

\[ x + 2(4) = 10 \]

\[ x + 8 = 10 \]

Subtracting 8 from both sides:

\[ x = 2 \]

### Step 9: Substitute \( x \) and \( z \) to find \( y \)

Now substitute \( x = 2 \) into the expression for \( y \):

\[ y = x - 3 = 2 - 3 = -1 \]

### Final Values

Thus, the solution to the system of equations is:

\[ x = 2, \quad y = -1, \quad z = 4 \]