To find the solution set of the system of equations:
[tex]\[
\left\{
\begin{array}{l}
2x - y = 5 \\
3y + 15 = 6x
\end{array}
\right.
\][/tex]
we will solve this system step-by-step.
### Step 1: Solve the first equation for [tex]\( y \)[/tex]
The first equation is:
[tex]\[
2x - y = 5
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = 2x - 5
\][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] in the second equation
We now have [tex]\( y = 2x - 5 \)[/tex]. Substitute this into the second equation:
[tex]\[
3(2x - 5) + 15 = 6x
\][/tex]
### Step 3: Simplify the equation
Expand and simplify:
[tex]\[
6x - 15 + 15 = 6x
\][/tex]
[tex]\[
6x = 6x
\][/tex]
This is an identity, meaning it is true for all [tex]\( x \in \mathbb{R} \)[/tex].
### Step 4: Conclusion
Since the second equation holds true for any [tex]\( x \)[/tex], there are infinitely many solutions. The solution set is determined by the relation [tex]\( y = 2x - 5 \)[/tex].
### Step 5: Define the solution set
Therefore, the solution set can be written as follows:
[tex]\[
\{(x, y) \mid y = 2x - 5, x \in \mathbb{R}\}
\][/tex]
In the context of the provided answer choices, this matches with:
C. [tex]\(\{(t, 2t - 5) \mid t \in \mathbb{R}\}\)[/tex]
Thus, the answer is:
C. [tex]\(\{(t, 2t - 5) \mid t \in \mathbb{R}\}\)[/tex]
