To solve the system of equations:
[tex]\[
\begin{cases}
x = 2y - 5 \\
-3x = -6y + 15
\end{cases}
\][/tex]
we will go through the following steps:
1. Let's use the first equation to express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[
x = 2y - 5
\][/tex]
2. Next, substitute this expression for [tex]\(x\)[/tex] into the second equation:
[tex]\[
-3(2y - 5) = -6y + 15
\][/tex]
3. Distribute [tex]\(-3\)[/tex] on the left-hand side:
[tex]\[
-6y + 15 = -6y + 15
\][/tex]
4. Notice that the equation simplifies to a true statement:
[tex]\[
-6y + 15 = -6y + 15
\][/tex]
This simplification shows that the second equation is identical to the first equation in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. It indicates that both equations represent the same line. Therefore, rather than having a unique solution, every point [tex]\((x, y)\)[/tex] on this line satisfies both equations.
Given that this system of equations represents the same line, there are infinitely many solutions.
Hence, the correct answer is:
d. Infinite solutions
