Sure, let's work through solving this step-by-step.
### 1. Understand the System of Equations
We have the following system of linear equations:
[tex]\[ \begin{array}{l}
x + y = 5 \\
x + y = 3
\end{array} \][/tex]
### 2. Graphing the Equations
To graph these equations, let's first understand what each of them represents.
#### Equation 1: [tex]\( x + y = 5 \)[/tex]
- If [tex]\( x = 0 \)[/tex], then [tex]\( y = 5 \)[/tex]. So one point on the graph is [tex]\( (0, 5) \)[/tex].
- If [tex]\( y = 0 \)[/tex], then [tex]\( x = 5 \)[/tex]. So another point on the graph is [tex]\( (5, 0) \)[/tex].
We can draw a straight line passing through these two points.
#### Equation 2: [tex]\( x + y = 3 \)[/tex]
- If [tex]\( x = 0 \)[/tex], then [tex]\( y = 3 \)[/tex]. So one point on the graph is [tex]\( (0, 3) \)[/tex].
- If [tex]\( y = 0 \)[/tex], then [tex]\( x = 3 \)[/tex]. So another point on the graph is [tex]\( (3, 0) \)[/tex].
We can also draw a straight line passing through these two points.
### 3. Compare the Lines on the Graph
Notice that both lines have the same slope but different intercepts. They are parallel lines. Parallel lines will never intersect.
### 4. Identify the Solution
The solution to a system of equations represented graphically corresponds to points where the lines intersect.
Since the lines [tex]\( x + y = 5 \)[/tex] and [tex]\( x + y = 3 \)[/tex] are parallel and do not intersect, there is no point that satisfies both equations simultaneously.
### Conclusion
Hence, the system of equations:
[tex]\[ \begin{array}{l}
x + y = 5 \\
x + y = 3
\end{array} \][/tex]
has no solution.
So, the correct option is:
- No Solution
