A linear function can be expressed in the form of \( y = mx + b \), where \( m \) and \( b \) are constants. This means that the function must involve no exponents other than 1 for the variable \( x \), and all terms can only be at most linear.
Now let's analyze each of the given options:
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\( y = -12x + 78 \): This is in the form of \( y = mx + b \) where \( m = -12 \) and \( b = 78 \). This is a linear function.
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\( x^2 + y = 33 \): This can be rearranged to \( y = -x^2 + 33 \). Because of the \( x^2 \) term, it's a quadratic function, not a linear function.
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\( 2y - 5x = 9 \): Rearranging this gives \( y = \frac{5}{2}x + \frac{9}{2} \), which is also in the form \( y = mx + b \). This is a linear function.
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\( y = 2x^3 + 4 \): The presence of the \( x^3 \) term indicates that this is a cubic function, not a linear function.
Based on this analysis, the functions that are linear are:
- \( y = -12x + 78 \)
- \( 2y - 5x = 9 \) (which can be rearranged to the linear form)
So the correct options for linear functions are:
- \( y = -12x + 78 \)
- \( 2y - 5x = 9 \) (after rearranging)
However, since you've indicated only one can be selected in a response, the first one listed (\( y = -12x + 78 \)) is a straightforward choice.