Solve: 
2x + y + z = 36
(2y + x) - z = 27
y + 5x = 24

We have a system of three linear equations:

1. \( 2x + y + z = 36 \)
2. \( (2y + x) - z = 27 \)
3. \( y + 5x = 24 \)

We will solve this system step by step using substitution or elimination.

### Step 1: Express \( y \) in terms of \( x \) from the third equation.

From the third equation \( y + 5x = 24 \), solve for \( y \):

\[
y = 24 - 5x
\]

### Step 2: Substitute \( y = 24 - 5x \) into the first and second equations.

#### Substitute into the first equation:

\[
2x + (24 - 5x) + z = 36
\]

Simplify:

\[
2x + 24 - 5x + z = 36
\]
\[
-3x + z = 12 \quad \text{(Equation A)}
\]
\[
z = 3x + 12
\]

#### Substitute into the second equation:

\[
(2(24 - 5x) + x) - z = 27
\]
Simplify:

\[
48 - 10x + x - z = 27
\]
\[
-9x + 48 - z = 27
\]
Substitute \( z = 3x + 12 \) from Equation A:

\[
-9x + 48 - (3x + 12) = 27
\]
Simplify:

\[
-9x + 48 - 3x - 12 = 27
\]
\[
-12x + 36 = 27
\]
\[
-12x = -9
\]
\[
x = \frac{3}{4}
\]

### Step 3: Substitute \( x = \frac{3}{4} \) into \( y = 24 - 5x \) and \( z = 3x + 12 \).

#### For \( y \):

\[
y = 24 - 5\left(\frac{3}{4}\right)
\]
\[
y = 24 - \frac{15}{4} = \frac{96}{4} - \frac{15}{4} = \frac{81}{4}
\]

#### For \( z \):

\[
z = 3\left(\frac{3}{4}\right) + 12
\]
\[
z = \frac{9}{4} + 12 = \frac{9}{4} + \frac{48}{4} = \frac{57}{4}
\]

### Final solution:
\[
x = \frac{3}{4}, \quad y = \frac{81}{4}, \quad z = \frac{57}{4}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve systems of linear equations using substitution?
2. Can this system be solved graphically?
3. What methods are there to solve larger systems of equations?
4. How can matrix algebra be used to solve systems of equations?
5. How do you check if a system of equations has no solution?

**Tip:** In a system of linear equations, substitution works well when one of the equations is easy to solve for a variable.

Solve: 2x + y + z = 36, (2y + x) - z = 27, y + 5x = 24

Learn how to solve the system of linear equations 2x + y + z = 36, (2y + x) - z = 27, y + 5x = 24 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the substitution and elimination methods, and is suitable for students in Grades 9-12.

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