To solve the system of equations:
[tex]\[
\begin{array}{c}
x + y = 3 \\
x - 3y = -9
\end{array}
\][/tex]
we will use the method of substitution or elimination. Let's use the elimination method:
Step 1: Write both equations in standard form.
The equations are already in standard form:
1. [tex]\(x + y = 3\)[/tex]
2. [tex]\(x - 3y = -9\)[/tex]
Step 2: Eliminate one of the variables.
We can eliminate [tex]\(x\)[/tex] by subtracting the second equation from the first equation. To align the coefficients of [tex]\(x\)[/tex], we don't need to do anything since both equations already have [tex]\(x\)[/tex] with coefficient 1 in the first equation and -1 in the second equation:
[tex]\[
\begin{array}{c}
x + y = 3 \\
(-(x - 3y) = -(-9)) \Rightarrow -x + 3y = 9
\end{array}
\][/tex]
Now add the two equations:
[tex]\[
(x + y) + (-x + 3y) = 3 + 9
\][/tex]
This simplifies to:
[tex]\[
4y = 12
\][/tex]
Step 3: Solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{12}{4} = 3
\][/tex]
Step 4: Substitute [tex]\(y = 3\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex].
Using the first equation [tex]\(x + y = 3\)[/tex]:
[tex]\[
x + 3 = 3
\][/tex]
[tex]\[
x = 3 - 3
\][/tex]
[tex]\[
x = 0
\][/tex]
Thus, the solution to the system of equations is [tex]\((x, y) = (0, 3)\)[/tex].
Step 5: Verify the solution:
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 3\)[/tex] into the second equation:
[tex]\[
0 - 3 \cdot 3 = -9
\][/tex]
[tex]\[
0 - 9 = -9
\][/tex]
This confirms that our solution is correct.
Comparing this with the given options:
(A) [tex]\((-3, 0)\)[/tex]
(B) [tex]\((0, 3)\)[/tex]
(C) [tex]\((1, 2)\)[/tex]
(D) [tex]\((9, -6)\)[/tex]
The correct answer is:
[tex]\(\boxed{(0, 3)}\)[/tex], which corresponds to option (B).
