To determine which system of equations could lead to the intermediate step of \(9x = 27\) during elimination, we can manipulate that equation and consider the context that led to it.
- We can rewrite \(9x = 27\) as \(x = \frac{27}{9} = 3\).
- This means that after some steps in the elimination process, we isolate \(x\) and find that \(x = 3\).
Now, for elimination to reach this point, we need to consider combinations of two linear equations that can be manipulated to this form.
We start with two linear equations in the form:
- Equation 1: \(a_1x + b_1y = c_1\)
- Equation 2: \(a_2x + b_2y = c_2\)
Through the elimination process, we want to eliminate \(y\), leading us to an equation with \(x\).
For it to result in \(9x = 27\), we can try finding pairs of equations where multiplying (or combining terms) can lead to a scenario where we arrive at \(9x = 27\).
Example
Consider the following equations:
- \(3x + 2y = 12\) (Multiply this equation by 3 will give \(9x + 6y = 36\))
- \(3x - 2y = 6\) (Adding these equations leads to elimination of \(y\))
After adjusting the equations, we can see:
- Adding the two equations gives: \[ (3x + 2y) + (3x - 2y) = 12 + 6 \implies 6x = 18 \]
- Multiplying the first equation by 3 gives the contribution towards the \(9x\): \[ 9x + 6y = 36 \]
- To isolate \(x\) you could also use forms that would align with the elimination process, eventually leading us to the point of elimination producing \(9x = 27\).
Conclusion
So, to construct a system of equations that leads to the intermediate step of \(9x = 27\), we can consider systems where by eliminating \(y\) (through adding or subtracting the equations appropriately), we can isolate \(x\) to form that equation.
A valid example of the equations would be any two linear equations in \(x\) and \(y\) whose combination will yield this result.
If you're provided with options, the one whose manipulation leads to \(9x = 27\) (or any rearrangement thereof) would be the correct choice.
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