2. Without using a graphing calculator, identify all of the following equations that straight lines. Explain how you decide. $$ \begin{array}{ll} ext { a. } y=3 x & ext { e. } y=\frac{1}{x}+3\end{array} $$

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2. Without using a graphing calculator, identify all of the following equations that straight lines. Explain how you decide. $$ \begin{array}{ll}	ext { a. } y=3 x & 	ext { e. } y=\frac{1}{x}+3\end{array} $$

UpStudy AI · Accepted Answer

To determine which of the given equations represent straight lines, we need to analyze their forms. A straight line can be represented in the slope-intercept form, which is $$ y = mx + b $$, where $$ m $$ is the slope and $$ b $$ is the y-intercept.

Let's examine each equation:

### a. $$ y = 3x $$

This equation is already in the slope-intercept form:
- Here, $$ m = 3 $$ (the slope) and $$ b = 0 $$ (the y-intercept).
- Since it can be expressed as $$ y = mx + b $$, this equation represents a straight line.

### e. $$ y = \frac{1}{x} + 3 $$

This equation is not in the slope-intercept form. To analyze it further:
- The term $$ \frac{1}{x} $$ indicates that this equation is not linear. Instead, it represents a hyperbola, as it involves a variable in the denominator.
- The presence of $$ \frac{1}{x} $$ means that as $$ x $$ approaches zero, $$ y $$ approaches infinity, and as $$ x $$ increases or decreases, $$ y $$ will not change linearly.

### Conclusion

- **Equation a** $$ y = 3x $$ is a straight line.
- **Equation e** $$ y = \frac{1}{x} + 3 $$ is not a straight line.

Thus, the only equation that represents a straight line is **a. $$ y = 3x $$**.
Only equation a. $$ y = 3x $$ represents a straight line.

2. Without using a graphing calculator, identify all of the following equations that straight lines. Explain how you decide. \( \begin{array}{ll}\text { a. } y=3 x & \text { e. } y=\frac{1}{x}+3\end{array} \)

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