Sure! Let's solve the given system of equations using the substitution method.
The system of equations is:
[tex]\[ 3x + 2y = 7 \][/tex]
[tex]\[ x = 3y + 6 \][/tex]
Step 1: Substitute the expression for [tex]\( x \)[/tex] from the second equation into the first equation.
The second equation gives us:
[tex]\[ x = 3y + 6 \][/tex]
We substitute this expression for [tex]\( x \)[/tex] into the first equation:
[tex]\[ 3(3y + 6) + 2y = 7 \][/tex]
Step 2: Simplify the equation.
First, distribute 3 in the equation:
[tex]\[ 9y + 18 + 2y = 7 \][/tex]
Next, combine like terms:
[tex]\[ 11y + 18 = 7 \][/tex]
Step 3: Solve for [tex]\( y \)[/tex].
Subtract 18 from both sides of the equation:
[tex]\[ 11y = 7 - 18 \][/tex]
[tex]\[ 11y = -11 \][/tex]
Divide both sides by 11:
[tex]\[ y = -1 \][/tex]
Step 4: Substitute the value of [tex]\( y \)[/tex] back into the second equation to find [tex]\( x \)[/tex].
We already have the equation:
[tex]\[ x = 3y + 6 \][/tex]
Substitute [tex]\( y = -1 \)[/tex] into the equation:
[tex]\[ x = 3(-1) + 6 \][/tex]
[tex]\[ x = -3 + 6 \][/tex]
[tex]\[ x = 3 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ (x, y) = (3, -1) \][/tex]
Out of the given options:
[tex]\[
\begin{array}{l}
(0, -2) \\
(1, 2) \\
(3, -1) \\
(6, 0)
\end{array}
\][/tex]
The correct solution is:
[tex]\[ (3, -1) \][/tex]
