To eliminate the \(x\) terms from two equations by adding them together, you need to find a common coefficient for the \(x\) terms. Let's denote the two equations generically as:
- \(a_1 x + b_1 y = c_1\)
- \(a_2 x + b_2 y = c_2\)
To eliminate \(x\), we can multiply the first equation by a constant that will make the coefficient of \(x\) equal in magnitude (but opposite in sign) to the coefficient of \(x\) in the second equation.
Let's analyze the provided options by looking for the two pairs of multipliers \(m_1\) and \(m_2\):
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The first option is to multiply the first equation by 3 and the second equation by -5. If we let the coefficient of \(x\) in the first equation be \(a_1\), and in the second equation \(a_2\), we have:
- \(3a_1\) and \(-5a_2\)
In this case, for them to be equal in magnitude for cancellation, you need: \[3a_1 = -(-5a_2)\] or \[3a_1 = 5a_2\]
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The second option suggests multiplying the first equation by 3 and the second by 5:
- \(3a_1\) and \(5a_2\).
This would imply: \[3a_1 + 5a_2 = 0\] (no cancellation).
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The third option is to multiply the first equation by 7 and the second by -6:
- \(7a_1\) and \(-6a_2\)
For these to cancel: \[7a_1 = 6a_2\]
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The fourth option is to multiply the first equation by 7 and the second by 6:
- \(7a_1\) and \(6a_2\)
This would imply: \[7a_1 + 6a_2 = 0\] (no cancellation).
To determine the correct multipliers, the values must be given (the coefficients of \(x\) in each linear equation). However, without those actual coefficients, we cannot specifically validate which option is correct. Based on how you've presented the information:
If you want to eliminate \(x\), you typically want either to multiply by the opposite sign or look for pairs that can equalize.
Assuming we have no specific coefficients provided, logically you'd evaluate common multiples and cancellation techniques per the layout. This typically results in engaging logical pairs of common factor multiples to allow for additive elimination.
Ultimately, if the context of \(a_1\) and \(a_2\) reinforces multiplicative relations (like 3 and -5 or 7 and -6), you pursue that ratio setting or modal alignment.
If presented with more specific coefficients, we can clarify further the most viable option to hit the \(y\) solution efficiently. Given what's outlined, based only on numeric pairs (multipliers), you would seek to engage logical cancellation partner pairs (notably both positive/negative sign control) per above testing. If you can clarify the \(x\) term coefficients, that might help to detail the strategic path!