Question

To determine if the linear functions are the same, we compare their slopes and y-intercepts.

For the equation
:

Slope: -1
Y-intercept: 3
For the table, we can calculate the slope using any two points. Taking points
and
:

Slope =


Now, we find the y-intercept by plugging in
. Between the given points, it appears that when
,
is not directly available. However, we can check the calculated points: Using the slope and one point, we use point-slope form:


The y-intercept from the calculations is -3. Thus, the two functions differ in their y-intercepts (3 vs. -3) but share the same slope (-1). Therefore, these are not the same functions.

Answer 1

To compare the linear functions you've mentioned, let's summarize the analysis step by step.

1. **Equation Analysis**:
- The equation has a **slope of -1** and a **y-intercept of 3**. This means the equation can be expressed in slope-intercept form as:
\[
y = -1x + 3
\]

2. **Table Analysis**:
- You mentioned using points from a table to calculate the slope. Let's assume you took two points, say \( (x_1, y_1) \) and \( (x_2, y_2) \). To calculate the slope \( m \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
- If this calculation yields a slope of **-1**, then we can conclude that both functions share the same slope.

3. **Finding the Y-Intercept**:
- If you do not have the y-intercept directly from the table, you can use one of the points and the slope to find it using the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
- Rearranging this will allow you to solve for \( y \) when \( x = 0 \) (finding the y-intercept).

4. **Y-Intercept Comparison**:
- You found that the y-intercept from the table calculations is **-3**, while the y-intercept from the equation is **3**.

5. **Conclusion**:
- Both functions have the same slope of **-1**, but the y-intercepts are different (3 vs. -3). Therefore, the functions are not the same.

In summary, while they share the same slope, the differing y-intercepts confirm that these are indeed two distinct linear functions.