Question

Which equation models the relationship between the two variables in the table below?

X. y
2. -4
3. -7
4. -10
5. -13

Answer 1

To find the equation that models the relationship between the two variables \( x \) and \( y \) in the table, we can look for a pattern in the values.

We start by listing the given pairs:

- For \( x = 2 \), \( y = -4 \)
- For \( x = 3 \), \( y = -7 \)
- For \( x = 4 \), \( y = -10 \)
- For \( x = 5 \), \( y = -13 \)

Next, we examine how \( y \) changes as \( x \) changes:

- From \( x = 2 \) to \( x = 3 \) (increase of 1 in \( x \)): \( y \) changes from -4 to -7 (decrease of 3).
- From \( x = 3 \) to \( x = 4 \) (increase of 1 in \( x \)): \( y \) changes from -7 to -10 (decrease of 3).
- From \( x = 4 \) to \( x = 5 \) (increase of 1 in \( x \)): \( y \) changes from -10 to -13 (decrease of 3).

We see that for each increase of 1 in \( x \), \( y \) decreases by 3. This suggests a linear relationship.

The slope \( m \) can be calculated as:
\[
m = \frac{\Delta y}{\Delta x} = \frac{-3}{1} = -3
\]

Next, we can use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Using the point \( (2, -4) \) and the slope \( m = -3 \):
\[
y - (-4) = -3(x - 2)
\]
Simplifying gives:
\[
y + 4 = -3x + 6
\]
\[
y = -3x + 6 - 4
\]
\[
y = -3x + 2
\]

Thus, the equation that models the relationship between \( x \) and \( y \) is:
\[
y = -3x + 2
\]