Let's solve the given system of equations step-by-step and determine the relationship between the solution of the initial system and the equation [tex]\(3x + y = 7\)[/tex].
### Initial System of Equations
The system of equations provided is:
[tex]\[
\begin{cases}
x + y = 3 \quad \text{(1)} \\
x - y = 1 \quad \text{(2)}
\end{cases}
\][/tex]
### Step 1: Solving the System
First, we'll solve this system by adding the two equations to eliminate [tex]\(y\)[/tex].
1. Add equations (1) and (2):
[tex]\[
(x + y) + (x - y) = 3 + 1
\][/tex]
Simplifying the left side:
[tex]\[
x + y + x - y = 2x = 4
\][/tex]
So, we have:
[tex]\[
2x = 4 \implies x = 2
\][/tex]
2. Substitute [tex]\(x = 2\)[/tex] back into equation (1) to find [tex]\(y\)[/tex]:
[tex]\[
2 + y = 3 \implies y = 1
\][/tex]
### Step 2: Verify the Solution in the Second Equation
Let's verify our solution [tex]\((x, y) = (2, 1)\)[/tex] by substituting into equation (2):
[tex]\[
2 - 1 = 1 \quad \text{which is true}
\][/tex]
### Step 3: Check the Relationship with [tex]\(3x + y = 7\)[/tex]
Now, we need to determine if the solution [tex]\((2, 1)\)[/tex] also satisfies the equation [tex]\(3x + y = 7\)[/tex]:
Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex] into [tex]\(3x + y\)[/tex]:
[tex]\[
3(2) + 1 = 6 + 1 = 7
\][/tex]
This is true, meaning the solution [tex]\((2, 1)\)[/tex] satisfies the equation [tex]\(3x + y = 7\)[/tex].
### Conclusion
Given that our solution [tex]\((2, 1)\)[/tex] satisfies both the initial system of equations and the equation [tex]\(3x + y = 7\)[/tex], the correct statement is:
The solution of the initial system is also a solution of the equation [tex]\(3x + y = 7\)[/tex].
So, the answer is:
[tex]\[
\boxed{\text{The solution of the initial system is also a solution of the equation } 3x + y = 7.}
\][/tex]
