To solve the system of equations using the substitution method, we first need to express one variable in terms of the other from one of the equations. Let's start with the second equation:
\[ y + 6 = 2x \]
We can solve for \(y\):
\[ y = 2x - 6 \]
Now that we have \(y\) expressed in terms of \(x\), we can substitute this expression for \(y\) into the first equation:
\[ 9x - 3y = 12 \]
Substituting \(y = 2x - 6\) into the equation, we get:
\[ 9x - 3(2x - 6) = 12 \]
Now, distribute the \(-3\):
\[ 9x - 6x + 18 = 12 \]
Combine like terms:
\[ 3x + 18 = 12 \]
Next, subtract \(18\) from both sides:
\[ 3x = 12 - 18 \]
\[ 3x = -6 \]
Now, divide by \(3\):
\[ x = -2 \]
Now that we have the value of \(x\), we can substitute it back into the expression we found for \(y\):
\[ y = 2x - 6 \]
Substituting \(x = -2\):
\[ y = 2(-2) - 6 \]
\[ y = -4 - 6 \]
\[ y = -10 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (-2, -10) \]
To verify, we can substitute \(x = -2\) and \(y = -10\) back into the original equations:
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First equation: \[ 9(-2) - 3(-10) = 12 \] \[ -18 + 30 = 12 \quad \text{True} \]
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Second equation: \[ -10 + 6 = 2(-2) \] \[ -4 = -4 \quad \text{True} \]
Both equations are satisfied, confirming that the solution is correct.
The final answer is:
\[ (x, y) = (-2, -10) \]