To solve the system of equations, we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously. The system of equations given is:
[tex]\[
-3x + 6y = 24 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
-2x - y = 1 \quad \text{(Equation 2)}
\][/tex]
Let's use the method of elimination to solve this system step-by-step:
1. Equation Simplification:
First, we can simplify Equation 1 by dividing every term by 3:
[tex]\[
-3x + 6y = 24 \implies -x + 2y = 8
\][/tex]
Hence, the system of equations now looks like this:
[tex]\[
-x + 2y = 8 \quad \text{(Simplified Equation 1)}
\][/tex]
[tex]\[
-2x - y = 1 \quad \text{(Equation 2)}
\][/tex]
2. Elimination:
Now, let's eliminate one of the variables. We can start by multiplying Simplified Equation 1 by 2, to align it with the [tex]\( x \)[/tex]-terms in Equation 2:
[tex]\[
-2(-x + 2y) = -2 \cdot 8 \implies -2x + 4y = 16
\][/tex]
Subtract Equation 2 from this new equation:
[tex]\[
(-2x + 4y) - (-2x - y) = 16 - 1
\][/tex]
Simplifying the left-hand side, we get:
[tex]\[
-2x + 4y + 2x + y = 15
\][/tex]
This reduces to:
[tex]\[
5y = 15
\][/tex]
Solving for [tex]\( y \)[/tex], we divide both sides by 5:
[tex]\[
y = 3
\][/tex]
3. Substitute [tex]\( y \)[/tex] back into one of the original equations:
Let's use Simplified Equation 1 to find [tex]\( x \)[/tex]:
[tex]\[
-x + 2y = 8
\][/tex]
Substitute [tex]\( y = 3 \)[/tex]:
[tex]\[
-x + 2(3) = 8
\][/tex]
Simplify:
[tex]\[
-x + 6 = 8
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
-x = 8 - 6
\][/tex]
[tex]\[
-x = 2
\][/tex]
[tex]\[
x = -2
\][/tex]
4. Solution:
The solution to the system of equations is:
[tex]\[
(x, y) = (-2, 3)
\][/tex]
Thus, the correct answer is [tex]\((-2, 3)\)[/tex].
