To solve the system of equations using substitution, we’ll follow these steps:
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Identify the equations: \[ \text{Equation 1: } y = -x + 5 \] \[ \text{Equation 2: } 3x + y = 11 \]
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Substitute Equation 1 into Equation 2: Since we have \(y\) expressed in terms of \(x\) in Equation 1, we can substitute \(-x + 5\) for \(y\) in Equation 2.
So, substituting into Equation 2: \[ 3x + (-x + 5) = 11 \]
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Simplify the equation: Combine like terms: \[ 3x - x + 5 = 11 \] \[ 2x + 5 = 11 \]
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Solve for \(x\): Subtract 5 from both sides: \[ 2x = 11 - 5 \] \[ 2x = 6 \]
Now divide by 2: \[ x = \frac{6}{2} \] \[ x = 3 \]
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Substitute back to find \(y\): Now that we have \(x\), substitute \(3\) back into Equation 1 to find \(y\): \[ y = -x + 5 \] \[ y = -3 + 5 \] \[ y = 2 \]
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Solution: The solution to the system of equations is: \[ (x, y) = (3, 2) \]
So, the final answer is: \[ \boxed{(3, 2)} \]
Let’s verify the solution by plugging \(x = 3\) and \(y = 2\) back into both original equations.
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Checking Equation 1: \[ y = -x + 5 \] \[ 2 = -3 + 5 \quad \text{(True)} \]
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Checking Equation 2: \[ 3x + y = 11 \] \[ 3(3) + 2 = 9 + 2 = 11 \quad \text{(True)} \]
The solution \((3, 2)\) satisfies both equations. Thus, it is correct.