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

Question

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.