Solve the system of equations by elimination.
5x+5y–z=11
4x–2y+2z=24
–
3x–3y–z=
–
5

Let's solve the system of equations using the elimination method.

The system of equations is:
\[
5x + 5y - z = 11 \quad \text{(1)}
\]
\[
4x - 2y + 2z = 24 \quad \text{(2)}
\]
\[
3x - 3y - z = -5 \quad \text{(3)}
\]

### Step 1: Eliminate \( z \)

We'll start by eliminating \( z \) from two pairs of equations.

#### Pair 1: Equation (1) and Equation (3)
\[
(5x + 5y - z) - (3x - 3y - z) = 11 - (-5)
\]
Simplify:
\[
5x + 5y - z - (3x - 3y - z) = 11 + 5
\]
\[
(5x - 3x) + (5y + 3y) = 16
\]
\[
2x + 8y = 16 \quad \text{(4)}
\]

#### Pair 2: Equation (2) and Equation (3)
\[
(4x - 2y + 2z) - (3x - 3y - z) = 24 - (-5)
\]
Simplify:
\[
4x - 2y + 2z - (3x - 3y - z) = 24 + 5
\]
\[
(4x - 3x) + (-2y + 3y) + (2z + z) = 29
\]
\[
x + y + 3z = 29 \quad \text{(5)}
\]

### Step 2: Eliminate \( x \) and \( y \)

Now, let's solve the new system of equations (4) and (5):
\[
2x + 8y = 16 \quad \text{(4)}
\]
\[
x + y + 3z = 29 \quad \text{(5)}
\]

We can solve for one variable, say \( x \), in terms of \( y \) from equation (4):
\[
x = \frac{16 - 8y}{2} = 8 - 4y
\]
Now, substitute this expression for \( x \) into equation (5):
\[
(8 - 4y) + y + 3z = 29
\]
Simplify:
\[
8 - 3y + 3z = 29
\]
\[
3z - 3y = 21
\]
\[
z - y = 7 \quad \text{(6)}
\]

### Step 3: Substitute back into the system

From equation (6), we have:
\[
z = y + 7
\]
Substitute \( z = y + 7 \) into equation (5):
\[
x + y + 3(y + 7) = 29
\]
\[
x + y + 3y + 21 = 29
\]
\[
x + 4y = 8 \quad \text{(7)}
\]

Now, we substitute \( x = 8 - 4y \) into equation (7):
\[
(8 - 4y) + 4y = 8
\]
\[
8 = 8
\]
This is an identity, which means the system has **infinitely many solutions**.

The solution is a parametric system where:
\[
x = 8 - 4y
\]
\[
z = y + 7
\]
where \( y \) can be any real number.

Would you like more details or have any questions? Here are 5 related questions:

1. What is the difference between elimination and substitution methods?
2. Can this system be solved graphically, and what would the solution look like?
3. How do you interpret the result of infinitely many solutions in a real-world context?
4. How can you check the solution by substituting values for \( y \)?
5. What if you added an additional constraint, would the system have a unique solution?

**Tip:** When systems of equations lead to an identity (like \( 8 = 8 \)), it indicates infinitely many solutions!

Solve the system of equations by elimination.
5x + 5y - z = 11
4x - 2y + 2z = 24
3x - 3y - z = -5

This detailed guide shows how to solve the system of equations 5x + 5y - z = 11, 4x - 2y + 2z = 24, and 3x - 3y - z = -5 using the elimination method. Explore key algebraic concepts, including solving systems of equations and understanding cases of infinitely many solutions. Ideal for Grades 9-12 students.

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