To solve the system of equations,
[tex]\[
\begin{array}{r}
y-15=3x \quad \text{(1)} \\
-2x+5y=-3 \quad \text{(2)}
\end{array}
\][/tex]
we will follow a systematic approach.
First, solve Equation (1) for [tex]\( y \)[/tex]:
[tex]\[
y - 15 = 3x
\][/tex]
Rearrange this to express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 3x + 15
\][/tex]
Next, substitute [tex]\( y = 3x + 15 \)[/tex] into Equation (2):
[tex]\[
-2x + 5(3x + 15) = -3
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
-2x + 15x + 75 = -3
\][/tex]
Combine like terms:
[tex]\[
13x + 75 = -3
\][/tex]
Subtract 75 from both sides:
[tex]\[
13x = -3 - 75
\][/tex]
[tex]\[
13x = -78
\][/tex]
Divide by 13:
[tex]\[
x = \frac{-78}{13}
\][/tex]
[tex]\[
x = -6
\][/tex]
Next, substitute [tex]\( x = -6 \)[/tex] back into the equation [tex]\( y = 3x + 15 \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
y = 3(-6) + 15
\][/tex]
[tex]\[
y = -18 + 15
\][/tex]
[tex]\[
y = -3
\][/tex]
Thus, the solution to the system of equations is [tex]\( (x, y) = (-6, -3) \)[/tex].
Among the given options, the correct answer is:
C. [tex]\((-6, -3)\)[/tex]
