Anonymous
3 days ago
Given that \( a = x + y \) or \( a = 2x - d \) and \( x = d + y \), find the possible outcomes of \( a \).
Mathematics
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ChatGPT
3 days ago
To find the possible outcomes of \( a \), we can substitute \( x = d + y \) into the equations \( a = x + y \) and \( a = 2x - d \).
Substitute \( x = d + y \) into \( a = x + y \):
\( a = (d + y) + y \)
\( a = d + 2y \)
Substitute \( x = d + y \) into \( a = 2x - d \):
\( a = 2(d + y) - d \)
\( a = 2d + 2y - d \)
\( a = d + 2y \)
Therefore, the possible outcomes of \( a \) are \( a = d + 2y \) and \( a = d + 2y \). This means that \( a \) can take on any value of the form \( d + 2y \), where \( d \) and \( y \) are any real numbers.
Substitute \( x = d + y \) into \( a = x + y \):
\( a = (d + y) + y \)
\( a = d + 2y \)
Substitute \( x = d + y \) into \( a = 2x - d \):
\( a = 2(d + y) - d \)
\( a = 2d + 2y - d \)
\( a = d + 2y \)
Therefore, the possible outcomes of \( a \) are \( a = d + 2y \) and \( a = d + 2y \). This means that \( a \) can take on any value of the form \( d + 2y \), where \( d \) and \( y \) are any real numbers.
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Google Gemini AI
3 days ago
From the second equation, we have \( x = d y \). Substituting this into the first equation, we get
$$a = x y = d y \cdot y = d y^2.$$
From the second equation, we also have \( x = \frac{a+d}{2} \). Substituting this into the first equation, we get
$$a = x y = \frac{a+d}{2} \cdot y = \frac{ay+dy}{2} = \frac{a+d}{2} \cdot y.$$
Solving for \( a \), we get
$$a = \frac{a+d}{2} \Rightarrow a = d.$$
Therefore, the possible outcomes of \( a \) are \( a = d \) and \( a = d y^2 \).
$$a = x y = d y \cdot y = d y^2.$$
From the second equation, we also have \( x = \frac{a+d}{2} \). Substituting this into the first equation, we get
$$a = x y = \frac{a+d}{2} \cdot y = \frac{ay+dy}{2} = \frac{a+d}{2} \cdot y.$$
Solving for \( a \), we get
$$a = \frac{a+d}{2} \Rightarrow a = d.$$
Therefore, the possible outcomes of \( a \) are \( a = d \) and \( a = d y^2 \).
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