To solve the system of equations using substitution, we will first solve one of the equations for one variable and then substitute that expression into the other equation.
Starting with the first equation:
\[ 2x + y = 8 \]
We can solve for \( y \):
\[ y = 8 - 2x \]
Now we will substitute this expression for \( y \) into the second equation:
\[ 6x - 7y = -16 \]
Substituting \( y \):
\[ 6x - 7(8 - 2x) = -16 \]
Now, we will simplify the equation:
\[ 6x - 56 + 14x = -16 \]
Combine like terms:
\[ 20x - 56 = -16 \]
Next, we will add 56 to both sides:
\[ 20x = 40 \]
Now, divide both sides by 20:
\[ x = 2 \]
Now that we have \( x \), we will substitute it back into the equation we used to solve for \( y \):
\[ y = 8 - 2x \]
Substituting \( x = 2 \):
\[ y = 8 - 2(2) \] \[ y = 8 - 4 \] \[ y = 4 \]
Now we have the values of \( x \) and \( y \):
\[ x = 2, \quad y = 4 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (2, 4) \]
You can check the solution by substituting \( x = 2 \) and \( y = 4 \) back into both original equations to verify that they hold true:
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For the first equation: \[ 2(2) + 4 = 4 + 4 = 8 \quad \text{(True)} \]
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For the second equation: \[ 6(2) - 7(4) = 12 - 28 = -16 \quad \text{(True)} \]
Both equations are satisfied, confirming that the solution is correct.
The final answer is:
\[ (x, y) = (2, 4) \]
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