To identify which of the given equations represents a linear function, we should look for equations that can be expressed in the form \( y = mx + b \), where \( m \) and \( b \) are constants.
Let's analyze each option:
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\( x^2 + y = 33 \): This can be rewritten as \( y = 33 - x^2 \), which is not linear due to the \( x^2 \) term.
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\( y = -12x + 78, x \neq 0 \): This is already in the form \( y = mx + b \) with \( m = -12 \) and \( b = 78\). This is a linear function.
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\( y = -\frac{12}{x} + \frac{7}{8}, x \neq 0 \): This is not linear; the term \(-\frac{12}{x}\) indicates a hyperbola.
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\( y = 2x^3 + 4 \): This contains the \( x^3 \) term, making it a cubic function, not linear.
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\( 2y - 5x = 9 \): We can rewrite this as \( y = \frac{5}{2}x + \frac{9}{2} \), which is in the linear form again.
Based on this analysis, the equations that represent linear functions are:
- \( y = -12x + 78 \)
- \( 2y - 5x = 9 \) (when rewritten as \( y = \frac{5}{2}x + \frac{9}{2} \))
Therefore, the correct response indicating a linear function is either \( y = -12x + 78 \) or \( 2y - 5x = 9 \).