Question

Which of the equations y=−1/2x+7/8, y=2x^3+4, 2y−5x=9, or x^2+y=33 represent linear functions?

Answer 1

To determine which of the given equations represent linear functions, we need to check if each equation can be expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

1. **Equation: \(y = -\frac{1}{2}x + \frac{7}{8}\)**

This equation is already in the form \(y = mx + b\) with \(m = -\frac{1}{2}\) and \(b = \frac{7}{8}\). Therefore, this represents a linear function.

2. **Equation: \(y = 2x^3 + 4\)**

This equation has a term \(2x^3\), which is not linear (the exponent of \(x\) is 3). Thus, this does not represent a linear function.

3. **Equation: \(2y - 5x = 9\)**

We can rearrange this equation into the slope-intercept form.
\[
2y = 5x + 9
\]
\[
y = \frac{5}{2}x + \frac{9}{2}
\]
This is in the form \(y = mx + b\) with \(m = \frac{5}{2}\) and \(b = \frac{9}{2}\). Therefore, this represents a linear function.

4. **Equation: \(x^2 + y = 33\)**

Rearranging this gives
\[
y = 33 - x^2
\]
The term \(-x^2\) indicates that this is a quadratic function (not linear). Thus, this does not represent a linear function.

**Conclusion:**
The equations that represent linear functions are:
- \(y = -\frac{1}{2}x + \frac{7}{8}\)
- \(2y - 5x = 9\) (after rearranging)

So the linear functions are the first and third equations.