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To find the correct equation for the line graphed, we can start by identifying two key characteristics of the line: the **slope** and the **y-intercept**.

1. **Y-Intercept**:
   The graph shows that the line crosses the y-axis at \( y = -1 \). This means the y-intercept, \( b \), is \( -1 \).

2. **Slope**:
   We can determine the slope \( m \) by choosing two points on the line. Let's use the points:
   - \( (0, -1) \), which is the y-intercept.
   - \( (3, 2) \), another point clearly on the line.
   
   The slope \( m \) is calculated using the slope formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substituting in the points \( (0, -1) \) and \( (3, 2) \):
   \[
   m = \frac{2 - (-1)}{3 - 0} = \frac{3}{3} = 1
   \]

So, the slope of the line is \( 1 \).

3. **Equation**:
   The equation of a line in slope-intercept form is \( y = mx + b \). Substituting \( m = 1 \) and \( b = -1 \), the equation becomes:
   \[
   y = x - 1
   \]

Therefore, the correct equation for the line graphed is \( y = x - 1 \).

Would you like further details on this, or do you have any questions?

Here are 5 related questions for further exploration:
1. How do you determine the slope when the line is vertical or horizontal?
2. What would the equation be if the y-intercept were different but the slope stayed the same?
3. How would the graph look if the slope were negative, say \( m = -1 \)?
4. Can you derive the equation of a line using different points?
5. How does the slope change if the points used to calculate it are far apart?

**Tip:** To quickly check a line's slope on a graph, remember that it represents the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Which choice is a correct equation for the line graphed below?

Learn how to find the correct linear equation for a graphed line. This problem focuses on identifying the slope and y-intercept to derive the equation y = x - 1. Suitable for students in Grades 8-10, this guide covers key algebraic concepts such as slope-intercept form and the slope formula.

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