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.