To analyze the given system of equations and verify the statements, let's first rewrite the equations in a more standard form:
- \(-x + 6y = 16\) (Equation 1)
- \(8x - 6y = -2\) (Equation 2)
Checking each statement:
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The x-variable will be eliminated when adding the system of equations.
- To check this, let's add the two equations. We can first rewrite Equation 1 in a form that makes it easier to add: \[ -x + 6y + 8x - 6y = 16 - 2 \] Simplifying this gives: \[ 7x = 14 \quad (\text{which means the x-variable is not eliminated}) \]
- This statement is false.
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The y-variable will be eliminated when adding the system of equations.
- From the same setup as above where we added the equations, we can see that the \(6y\) and \(-6y\) will cancel each other out: \[ -x + 6y + 8x - 6y = 16 - 2 \] This simplifies to: \[ 7x = 14 \quad (\text{the y-variable is eliminated}) \]
- This statement is true.
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The sum of the system of equations is Negative 8 x = 14.
- From the previous addition, we found \(7x = 14\), not \(-8x\).
- This statement is false.
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x = 2.
- Solving \(7x = 14\): \[ x = \frac{14}{7} = 2 \]
- Hence, this statement is true.
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y = 3.
- Now we can substitute \(x = 2\) back into one of the original equations to find \(y\). Using Equation 1: \[ -2 + 6y = 16 \rightarrow 6y = 18 \rightarrow y = 3 \]
- Hence, this statement is true.
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There is only one solution to the system of equations.
- Since we found a single solution \((x, y) = (2, 3)\) and the system appears to be consistent and independent (the equations represent two lines that intersect at one point), we conclude there is one solution.
- This statement is true.
Summary of True Statements:
- The y-variable will be eliminated when adding the system of equations.
- x = 2.
- y = 3.
- There is only one solution to the system of equations.
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