Certainly! To solve the system of equations using the substitution method, let's follow the steps:
Given system of equations:
1. [tex]\( x + 2y = 12 \)[/tex]
2. [tex]\( -x = -y - 6 \)[/tex]
First, let's simplify the second equation to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
-x = -y - 6
\][/tex]
By multiplying both sides of the equation by [tex]\(-1\)[/tex], we get:
[tex]\[
x = y + 6
\][/tex]
Now, we will substitute [tex]\( x = y + 6 \)[/tex] from the second equation into the first equation:
[tex]\[
(y + 6) + 2y = 12
\][/tex]
Combine like terms:
[tex]\[
y + 6 + 2y = 12
\][/tex]
[tex]\[
3y + 6 = 12
\][/tex]
Next, isolate [tex]\( y \)[/tex] by subtracting 6 from both sides:
[tex]\[
3y = 12 - 6
\][/tex]
[tex]\[
3y = 6
\][/tex]
Then, solve for [tex]\( y \)[/tex] by dividing both sides by 3:
[tex]\[
y = \frac{6}{3}
\][/tex]
[tex]\[
y = 2
\][/tex]
Now, we need to find the value of [tex]\( x \)[/tex] by substituting [tex]\( y = 2 \)[/tex] back into the equation [tex]\( x = y + 6 \)[/tex]:
[tex]\[
x = 2 + 6
\][/tex]
[tex]\[
x = 8
\][/tex]
Therefore, the solution to the system of equations is [tex]\((x, y) = (8, 2)\)[/tex].
The correct ordered pair is:
D. [tex]\((8, 2)\)[/tex]
