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Bot GPT 4-o mini To understand why the system of equations \( 7x + 7y = 42 \) and \( 6x + y = 36 \) is equivalent to the system of equations \( x + y = 6 \) and \( 6x + y = 36 \), we can analyze the first equation.
Starting with the equation \( 7x + 7y = 42 \), we can simplify it by dividing every term by 7:
\[
\frac{7x}{7} + \frac{7y}{7} = \frac{42}{7}
\]
This simplifies to:
\[
x + y = 6
\]
Now we have shown that \( 7x + 7y = 42 \) is equivalent to \( x + y = 6 \).
The second equation in both systems, \( 6x + y = 36 \), remains unchanged.
Thus, both systems actually contain the same linear equations, but expressed in different forms.
Among the provided options, the correct response is:
**The equation \( 7x + 7y = 42 \) can be divided by 7 to produce the equivalent equation \( x + y = 6 \).**