Question

determine if these are liner relationships or not

input output
x y
-2 -7
-1 -3
0 1
1 5
2 9
3 13

Answer 1

To determine if the relationship between the input (x) and output (y) values is linear, we can check if the difference between consecutive y-values is consistent for equal differences in x-values.

Let's calculate the differences in both x and y:

1. **Input Values (x)**: -2, -1, 0, 1, 2, 3
2. **Output Values (y)**: -7, -3, 1, 5, 9, 13

### Differences in x:
- From -2 to -1: \( -1 - (-2) = 1 \)
- From -1 to 0: \( 0 - (-1) = 1 \)
- From 0 to 1: \( 1 - 0 = 1 \)
- From 1 to 2: \( 2 - 1 = 1 \)
- From 2 to 3: \( 3 - 2 = 1 \)

### Differences in y:
- From -7 to -3: \( -3 - (-7) = 4 \)
- From -3 to 1: \( 1 - (-3) = 4 \)
- From 1 to 5: \( 5 - 1 = 4 \)
- From 5 to 9: \( 9 - 5 = 4 \)
- From 9 to 13: \( 13 - 9 = 4 \)

### Summary of Differences:
- The difference for x is consistently \( 1 \).
- The difference for y is consistently \( 4 \).

Since the change in y is consistent (4) for each equal change in x (1), the relationship between x and y is linear.

To find the equation of the linear relationship, we can use the slope-intercept form \( y = mx + b \).

1. **Calculating the slope (m)**:
\[
m = \frac{\Delta y}{\Delta x} = \frac{4}{1} = 4
\]

2. **Using one of the points to solve for b** (let's use the point (0, 1)):
\[
1 = 4(0) + b \implies b = 1
\]

### Final Equation:
The linear relationship can be expressed as:
\[
y = 4x + 1
\]

Thus, the relationship is linear.