We are given a system of three equations with three variables, $x$, $y$, and $z$. The equations are: $x + y = 4$ $z - y = 3$ $2x + 2z = 8$ We need to find the values of $x$, $y$, and $z$ that satisfy all three equations.

1. Problem Description

We are given a system of three equations with three variables, $x$, $y$, and $z$.

The equations are:

$x + y = 4$

$z - y = 3$

$2x + 2z = 8$

We need to find the values of $x$, $y$, and $z$ that satisfy all three equations.

2. Solution Steps

First, let's simplify the third equation by dividing both sides by 2:

$2x + 2z = 8$

$x + z = 4$

Now we have the following system of equations:

$x + y = 4$ (1)

$z - y = 3$ (2)

$x + z = 4$ (3)

From equation (1), we can express $y$ in terms of $x$:

$y = 4 - x$ (4)

From equation (3), we can express $z$ in terms of $x$:

$z = 4 - x$ (5)

Now substitute (4) and (5) into equation (2):

$z - y = 3$

$(4 - x) - (4 - x) = 3$

$4 - x - 4 + x = 3$

$0 = 3$

This is a contradiction. There must be an error in the problem statement, or there is no solution to this system of equations. However, let us try to find the relationship between the variables. From (1), $x + y = 4$. From (3), $x + z = 4$. Therefore, $x+y = x+z$, which implies $y=z$.

Substituting $y=z$ into (2), $z - y = 3$, we have $z - z = 3$, or $0=3$, which is a contradiction.

3. Final Answer

Since we have a contradiction, there is no solution to this system of equations.

No solution.