To solve the given system of linear equations, we will use the method of elimination or substitution. Here are the equations:
1. [tex]\(x - 2y = 14\)[/tex]
2. [tex]\(x + 3y = 9\)[/tex]
### Method 1: Substitution
First, let's solve one of the equations for [tex]\(x\)[/tex] or [tex]\(y\)[/tex]. We'll solve the second equation for [tex]\(x\)[/tex]:
[tex]\[ x + 3y = 9 \][/tex]
[tex]\[ x = 9 - 3y \][/tex]
Now, substitute this expression for [tex]\(x\)[/tex] into the first equation:
[tex]\[ (9 - 3y) - 2y = 14 \][/tex]
Simplify and solve for [tex]\(y\)[/tex]:
[tex]\[ 9 - 3y - 2y = 14 \][/tex]
[tex]\[ 9 - 5y = 14 \][/tex]
[tex]\[ -5y = 14 - 9 \][/tex]
[tex]\[ -5y = 5 \][/tex]
[tex]\[ y = -1 \][/tex]
Now that we have [tex]\(y = -1\)[/tex], substitute this value back into the expression for [tex]\(x\)[/tex]:
[tex]\[ x = 9 - 3(-1) \][/tex]
[tex]\[ x = 9 + 3 \][/tex]
[tex]\[ x = 12 \][/tex]
So, the solution to the system of equations is [tex]\(x = 12\)[/tex] and [tex]\(y = -1\)[/tex].
### Verification
To ensure our solution is correct, we can substitute [tex]\(x = 12\)[/tex] and [tex]\(y = -1\)[/tex] back into the original equations:
1. [tex]\( x - 2y = 14 \)[/tex]
[tex]\[ 12 - 2(-1) = 12 + 2 = 14 \][/tex]
This is true.
2. [tex]\( x + 3y = 9 \)[/tex]
[tex]\[ 12 + 3(-1) = 12 - 3 = 9 \][/tex]
This is also true.
Since both equations are satisfied with [tex]\(x = 12\)[/tex] and [tex]\(y = -1\)[/tex], the solution is correct.
### Solution
The solution to the system of equations is [tex]\(x = 12\)[/tex] and [tex]\(y = -1\)[/tex]. Therefore, the correct solution pair is:
[tex]\[
(12, -1)
\][/tex]
Thus, the answer is [tex]\((12, -1)\)[/tex].
