https://file.aiquickdraw.com/mathbot/file/172166237305598cueer1.jpg

The equations provided in the image are:

1. \( x + y + 2 = 6 \)
2. \( y + 2z = 8 \)

Let's solve this system step by step.

### Step 1: Simplify the equations
First, we simplify both equations to isolate the variables.

**Equation 1:**
\[ x + y + 2 = 6 \]
Subtract 2 from both sides:
\[ x + y = 4 \]

**Equation 2:**
\[ y + 2z = 8 \]

### Step 2: Express \( x \) in terms of \( y \)
From the simplified Equation 1:
\[ x = 4 - y \]

### Step 3: Express \( y \) in terms of \( z \)
From Equation 2:
\[ y = 8 - 2z \]

### Step 4: Substitute \( y \) in terms of \( z \) into the expression for \( x \)
Substitute \( y = 8 - 2z \) into \( x = 4 - y \):
\[ x = 4 - (8 - 2z) \]
Simplify:
\[ x = 4 - 8 + 2z \]
\[ x = -4 + 2z \]

### Solution
Thus, the solutions for the variables in terms of \( z \) are:
\[ x = 2z - 4 \]
\[ y = 8 - 2z \]

These represent the parametric solution to the system of equations. 

### Verifying the solutions
Let's verify the solutions by substituting \( x \) and \( y \) back into the original equations.

1. For \( x + y + 2 = 6 \):
\[ (2z - 4) + (8 - 2z) + 2 = 6 \]
Simplify:
\[ 2z - 4 + 8 - 2z + 2 = 6 \]
\[ 6 = 6 \]
This is true.

2. For \( y + 2z = 8 \):
\[ (8 - 2z) + 2z = 8 \]
Simplify:
\[ 8 = 8 \]
This is true.

The solutions are correct.

Do you want details or have any questions? Here are some questions you might want to ask:

1. How do you solve a system of linear equations using substitution?
2. What is the parametric form of a solution?
3. Can you explain how to check if the solutions are correct?
4. How do you solve a system of equations using the elimination method?
5. What are the possible methods to solve a system of equations?

**Tip:** Always verify your solutions by substituting them back into the original equations. This ensures that your solutions are correct.

Learn how to solve a system of linear equations using the substitution method. Step-by-step solution provided for equations involving multiple variables. Suitable for high school students.

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