To solve the system of equations using substitution, let’s follow a step-by-step approach:
First, let’s write down the given system of equations:
[tex]\[ 3x + 2y = 7 \][/tex]
[tex]\[ x = 3y + 6 \][/tex]
1. Substitute the value of [tex]\( x \)[/tex] from the second equation into the first equation.
Given [tex]\( x = 3y + 6 \)[/tex], substitute this into the first equation:
[tex]\[ 3(3y + 6) + 2y = 7 \][/tex]
2. Simplify the equation:
[tex]\[ 9y + 18 + 2y = 7 \][/tex]
3. Combine like terms:
[tex]\[ 11y + 18 = 7 \][/tex]
4. Isolate the term with [tex]\( y \)[/tex] by subtracting 18 from both sides of the equation:
[tex]\[ 11y = 7 - 18 \][/tex]
[tex]\[ 11y = -11 \][/tex]
5. Solve for [tex]\( y \)[/tex] by dividing both sides by 11:
[tex]\[ y = -1 \][/tex]
6. Now, substitute [tex]\( y = -1 \)[/tex] back into the second original equation to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 3(-1) + 6 \][/tex]
[tex]\[ x = -3 + 6 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the solution of the system of equations is:
[tex]\[ x = 3, \, y = -1 \][/tex]
7. Verify this solution with the potential choices given:
[tex]\[
(0, -2), (1, 2), (3, -1), (6, 0)
\][/tex]
The pair [tex]\((3, -1)\)[/tex] matches our solution.
Therefore, the correct choice is:
[tex]\[ (3, -1) \][/tex]
