To determine why the given system of equations is equivalent to the other system, we should analyze the first equation, \(7x + 7y = 42\).
We can simplify \(7x + 7y = 42\) by dividing all terms by 7:
\[ x + y = 6 \]
Thus, the first equation \(7x + 7y = 42\) is equivalent to \(x + y = 6\).
The second equation, \(6x + y = 36\), remains unchanged in both systems.
Therefore, the equivalent system of equations is:
- \(x + y = 6\)
- \(6x + y = 36\)
Now, let's evaluate the answers provided in your options:
- The equation 7x+7y=42 can be multiplied by 7 to produce the equivalent equation x+y=6. (This statement is incorrect; it should be divided by 7, not multiplied.)
- The intercepts of the equations 7x+7y=42 and x+y=6 are (0,6) and (6,0). (This is true, because when x=0, y=6, and when y=0, x=6.)
- The intercepts of the equations 7x+7y=42 and x+y=6 are (0,36) and (6,0). (This is incorrect as discussed.)
- All equations share the point (6,0). (This is not correct; \(x+y=6\) only shares the point at (6,0) while \(6x+y=36\) will not yield the same results from you will get different intercepts as discussed before.)
The most accurate statements related to the equivalency (based on the provided options) would be the one addressing the simplification through division rather than multiplication or focusing on the intercepts directly.
Given all this analysis, the correct response is:
- The equation \(7x+7y=42\) can be multiplied by 7 to produce the equivalent equation \(x+y=6\).
It should be noted that had it been correctly stated regarding division rather than multiplication in the above option, that would have been conclusively true. However, since it's ambiguous in the options, please verify the exact wording in those responses.